1. Introduction
Determining the minimum degree condition for the existence of spanning structures is a central problem in extremal graph theory. The first result of this direction is Dirac’s theorem [Reference Dirac7] in 1952 which states that for
$n\ge 3$
, every
$n$
-vertex graph
$G$
with
$\delta (G)\ge \frac{n}{2}$
contains a Hamiltonian cycle. Komlós et al. [Reference Komlós, Sárközy and Szemerédi24] proved that an
$n$
-vertex graph
$G$
with
$\delta (G)\geq (\frac{1}{2}+o(1))n$
contains a copy of every bounded-degree spanning tree, and in [Reference Komlós, Sárközy and Szemerédi25], the result is extended to trees with maximum degree
$O(\frac{n}{\log n})$
. Another notable extension is the bandwidth theorem of Böttcher et al. [Reference Böttcher, Schacht and Taraz4], which finds the (asymptotically) optimal minimum degree condition forcing spanning subgraphs with bounded chromatic number and sublinear bandwidth and resolves a conjecture of Bollobás and Komlós [Reference Komlós23].
Note that the extremal graphs in the above results usually have large independent sets, which makes them far from being typical. Hence a natural project is to study how the degree conditions drop if we forbid large independent sets from the host graph. Balogh et al. [Reference Balogh, Molla and Sharifzadeh1] initiated this study by proving if
$G$
is an
$n$
-vertex graph with
$\delta (G)\geq (\frac{1}{2}+o(1))n$
and
$\alpha (G)=o(n)$
, then
$G$
contains a
$K_{3}$
-factor. This result is asymptotically optimal and requires a weaker degree bound than
$\delta (G)\geq \frac{2}{3}n$
from the Corrádi–Hajnal theorem [Reference Corrádi and Hajnal6]. Nenadov and Pehova [Reference Nenadov and Pehova35] extended this result to the case of
$K_r$
-factor and asked for the best possible minimum degree condition on
$G$
with
$\alpha (G)=o(n)$
that guarantees a
$K_r$
-factor. Knierim and Su [Reference Knierim and Su22] resolved this question for
$r\geq 4$
by showing
$\delta (G)\geq (\frac{r-2}{r}+o(1))n$
is asymptotically best possible. Nenadov and Pehova [Reference Nenadov and Pehova35] also generalised
$\alpha (G)$
into
$\ell$
-independence number and this inspires several recent works [Reference Chang, Han, Kim, Wang and Yang5, Reference Han, Hu, Wang and Yang14, Reference Han, Morris, Wang and Yang15].
However, just excluding large independent sets is not enough to guarantee the existence of large connected subgraph. The union of two disjoint copies of
$K_{\frac{n}{2}}$
has independence number two, but it does not contain any connected subgraphs with more than
$\frac{n}{2}$
vertices. Hence it is necessary to impose stronger conditions to overcome this. The following notion of bipartite hole was introduced by McDiarmid and Yolov [Reference Mcdiarmid and Yolov33] in the study of Hamilton cycles. Given two disjoint vertex sets
$A$
and
$B$
in a graph
$G$
, we use
$E_{G}(A,B)$
to denote the set of edges joining
$A$
and
$B$
, and let
$e_{G}(A,B)=\vert E_{G}(A,B) \vert$
. An
$(s,t)$
-bipartite-hole in
$G$
consists of two disjoint sets
$S,\, T\subseteq V(G)$
with
$\vert S \vert =s$
and
$\vert T \vert =t$
such that
$e_{G}(S,T)=0$
. The bipartite-hole number
$\widetilde{\alpha }(G)$
refers to the maximum integer
$r$
such that
$G$
contains an
$(s,t)$
-bipartite-hole for every pair of nonnegative integers
$s$
and
$t$
with
$s+t=r$
. In this paper, we adopt a slightly different notion of bipartite-hole number due to Nenadov and Pehova [Reference Nenadov and Pehova35].
Definition 1.1. The bipartite independence number
$\alpha ^*(G)$
is the maximum integer
$t$
such that
$G$
contains a
$(t,t)$
-bipartite-hole.
It is clear from the definition that
$2\alpha ^*(G)+1\geq \widetilde{\alpha }(G)\geq \alpha ^*(G)+1$
. McDiarmid and Yolov [Reference Mcdiarmid and Yolov33] showed that
$\delta (G)\ge \widetilde{\alpha }(G)$
is enough to force
$G$
to be Hamiltonian and the minimum degree condition is sharp. Moreover, this was recently strengthened by Draganić, Correia, and Sudakov to the pancyclicity result [Reference Draganić, Correia and Sudakov8]. Also, Kim et al. [Reference Kim, Kim and Liu21] studied the decomposition of an almost regular graph
$G$
with
$\alpha ^*(G)=o(n)$
into almost spanning trees of bounded maximum degree.
The main result of this paper is the following. We denote by
$\mathcal{T}(n,\Delta )$
the family of all trees on
$n$
vertices with maximum degree at most
$\Delta$
.
Theorem 1.2.
For each
$ \varepsilon \gt 0$
and
$\Delta \in \mathbb{N}$
, there exists
$\alpha =\alpha (\varepsilon,\Delta )\gt 0$
such that the following holds for sufficiently large
$n \in \mathbb{N}$
. Every
$n$
-vertex graph
$G$
with
$\delta (G)\ge \varepsilon n$
and
$\alpha ^*(G)\lt \alpha n$
is
$\mathcal{T}(n,\Delta )$
-universal, that is,
$G$
contains every
$T\in \mathcal{T}(n,\Delta )$
as a subgraph.
That is, for a graph
$G$
with sublinear
$\alpha ^*(G)$
, the minimum degree condition forcing bounded-degree spanning trees is almost sublinear. We remark that the maximum degree
$\Delta$
of the tree in Theorem 1.2 cannot be larger than
$C\sqrt{n}$
for some constant
$C=C_{\alpha }\gt 0$
(in contrast to, for example, the result of [Reference Komlós, Sárközy and Szemerédi25], which holds for trees of maximum degree
$O(\frac{n}{\log n})$
) by the following construction. Given any
$\alpha \gt 0$
, choose positive integers
$n, k, \Delta, d$
such that
$k,\Delta$
are odd,
$n=\Delta k-k+2$
,
$\Delta \gt (k+2)d$
and
$\frac{1}{d}\ll \alpha$
. It is easy to see that
$\Delta \gt \sqrt{dn}$
. Let
$T$
be a caterpillar which consists of a path
$P=v_1v_2\ldots v_k$
with
$\Delta -1$
leaves attached to
$v_i$
for each
$i\in \{1,k\}$
and
$\Delta -2$
leaves attached to
$v_j$
for each
$j\in [2,k-1]$
. Now we are going to construct a graph
$G$
with
$\delta (G)=\frac{n}{2}+d-1$
and
$\alpha ^*(G)\le \alpha n$
, but that does not contain
$T$
as a subgraph. Let
$G_0=(V,E)$
be an
$(\frac{n}{2},d,\lambda )$
-regular graph such that
$\lambda \le \alpha d$
, whose existence is guaranteed by a result of Friedman [Reference Friedman11] on random
$d$
-regular graphs. Then we take two identical copies
$V_1$
and
$V_2$
of
$V$
, and join
$u\in V_1$
and
$v\in V_2$
if
$uv\in E(G_0)$
. The resulting bipartite
$d$
-regular graph is denoted by
$G^*$
. Note that by the well-known Expander Mixing Lemma (applied to
$G_0$
), we have that for any
$A\subseteq V_1,B\subseteq V_2$
each of size
$\alpha n$
,
\begin{equation*} e_{G^*}(A,B)=e_{G_0}(A,B)\ge \frac {d}{|G_0|}|A||B|-\lambda \sqrt {|A||B|}\ge \frac {2d}{n}(\alpha n)^2-(\alpha d)\alpha n\gt 0. \end{equation*}
We further add two disjoint copies of
$K_{n/2}$
on
$V_1$
,
$V_2$
and call the resulting graph
$G$
. Thus we have
$\alpha ^*(G)\le \alpha n$
. Suppose
$G$
contains a copy of
$T$
and
$V_1$
has at least
$\frac{k+1}{2}$
vertices of
$P$
since
$k$
is odd. Indeed, whenever we embed a branch vertex
$v_i\in V(P)$
into
$V_1$
, we have to embed at least
$\Delta -2-d$
leaves, which are attached to
$v_i$
, into
$V_1$
as well. Thus the order of
$V_1$
is at least
\begin{equation*}\frac {k+1}{2}+\frac {k+1}{2}(\Delta -2-d)=\frac {n+\Delta -(k+1)d-3}{2}\gt \frac {n}{2}\end{equation*}
as
$\Delta \gt (k+2)d$
, a contradiction.
Another motivation of our result is its connection to the randomly perturbed graphs introduced by Bohman et al. [Reference Bohman, Frieze and Martin2], where the host graph is obtained by adding random edges to a deterministic graph with minimum degree conditions. Krivelevich et al. [Reference Krivelevich, Kwan and Sudakov28] showed that for any
$\varepsilon \gt 0$
,
$\Delta \in \mathbb{N}$
and
$T\in \mathcal{T}(n,\Delta )$
, if
$G$
is an
$n$
-vertex graph with
$\delta (G)\geq \varepsilon n$
, then the randomly perturbed graph
$G\cup G(n,\frac{C}{n})$
a.a.s. contains
$T$
as a subgraph, where
$C$
depends only on
$\varepsilon$
and
$\Delta$
. They suggested that such a result can be improved to a universality result, that is,
$G\cup G(n,\frac{C}{n})$
a.a.s. contains all members of
$\mathcal{T}(n,\Delta )$
simultaneously. This is confirmed by Böttcher et al. [Reference Böttcher, Han, Kohayakawa, Montgomery, Parczyk and Person3]. Indeed, in their technical result which we state below, they replaced
$G(n,\frac{C}{n})$
by a deterministic sparse expander graph, and thus get the universality part for free.
Theorem 1.3 ([Reference Böttcher, Han, Kohayakawa, Montgomery, Parczyk and Person3], Theorem 2). For any
$\varepsilon \gt 0$
and integers
$C\ge 2$
and
$\Delta \gt 1$
, there exist
$\alpha \gt 0$
,
$D_0$
and
$n_0$
such that the following holds for any
$D\ge D_0$
and
$n\ge n_0$
. Let
$G$
be an
$n$
-vertex graph satisfying the following two conditions:
-
1.
$\Delta (G)\leq CD$
, -
2.
$e(U, W)\ge \frac{D}{Cn}|U||W|$
for all sets
$U, W\subseteq V(G)$
with
$|U|,|W|\ge \alpha n$
.
Suppose
$G_\varepsilon$
is an
$n$
-vertex graph on the same vertex set and
$\delta (G_\varepsilon )\ge \varepsilon n$
. Then
$H\;:\!=\;G_\varepsilon \cup G$
is
$\mathcal{T}(n,\Delta )$
-universal.
Note that the graph
$H$
in Theorem 1.3 satisfies
$\alpha ^*(H)\leq \alpha ^*(G)\lt \alpha n$
and
$\delta (H)\ge \delta (G_\varepsilon )\ge \varepsilon n$
, so
$H$
is
$\mathcal{T}(n,\Delta )$
-universal by Theorem 1.2. Hence Theorem 1.2 slightly improves upon Theorem 1.3, where we do not need to distinguish two graphs (or equivalently the maximum degree condition on the sparse graph
$G$
is no longer needed).
2. Proof strategy and preliminaries
2.1. Notation
For a graph
$G=(V,E)$
, let
$v(G)=\vert V\vert$
and
$e(G)=\vert E\vert$
. Given a collection of subgraphs
$\mathcal{F}=\left \{ F_{i}\;:\; i\in I\right \}$
, let
$V(\mathcal{F})=\cup _{i\in I}V(F_{i})$
. Given two vertex-disjoint graphs
$G_1, G_2$
, let
$G_1\cup G_2$
be the union of
$G_1$
and
$G_2$
. For
$U \subseteq V(G)$
, let
$G[U]$
be the induced subgraph of
$G$
on
$U$
and let
$G-U$
be the induced graph after removing
$U$
, that is
$G-U\;:\!=\;G[V\backslash U]$
. Given a vertex
$v\in V(G)$
and
$X,Y\subseteq V(G)$
, denoted by
$N_{X}(v)$
the set of neighbours of
$v$
in
$X$
and let
$d_{X}(v)\;:\!=\;\vert N_{X}(v)\vert$
. The neighbourhood of
$X$
in
$G$
is denoted by
$N_{G}(X)=(\cup _{v\in X}N(v))\backslash X$
and let
$N_{Y}(X)=N_{G}(X)\cap Y$
. We omit the index
$G$
if the graph is clear from the context.
For a graph
$F$
on
$[k]\;:\!=\;\left \{ 1,\ldots,k\right \}$
, we say that
$B$
is the
$n$
-blow-up of
$F$
if there exists a partition
$V_{1},\ldots,V_{k}$
of
$V(B)$
such that
$\vert V_{1}\vert =\cdots =\vert V_{k} \vert =n$
and we have that
$\{u,v\}\in E(B)$
if and only if
$u\in V_{i}$
and
$v\in V_{j}$
for some
$\{i,j\}\in E(F)$
. Given a spanning subgraph
$G$
of
$B$
, we call the sequence
$V_1,\dots, V_k$
the parts of
$G$
and we define
$\bar{\delta }(G)\;:\!=\;\min _{\{i,j\}\in E(F)}\delta (G[V_i,V_{j}])$
where
$G[V_i,V_{j}]$
is the bipartite subgraph of
$G$
induced by the parts
$V_i$
and
$V_{j}$
. A subset
$R$
of
$V(G)$
is balanced if
$\vert R\cap V_1\vert =\cdots =\vert R\cap V_k\vert$
. In particular, we say a subset of
$V(G)$
or a subgraph of
$G$
transversal if it intersects each part in exactly one vertex.
For a path
$P$
, the length of
$P$
is the number of edges in
$P$
. Given two vertices
$x,y$
, an
$x,y$
-path is a path with ends
$x$
and
$y$
. Let
$T$
be a tree and
$T^{\prime }$
be obtained from
$T$
by removing all leaves. A pendant star is a maximal star centred at a leaf of
$T^{\prime }$
, where the unique neighbour of the centre inside
$T^{\prime }$
is called the root of the pendant star. A bare path in
$T$
is a path whose internal vertices have degree exactly two in
$T$
. A caterpillar in
$T$
consists of a bare path in
$T^{\prime }$
as the central path with some (possibly empty) leaves attached to the internal vertices of the central path, where branch vertices are the internal vertices attached with at least one leaf. The length and ends of the caterpillar refer to the length and ends of its central path.
For two graphs
$H$
and
$G$
, an embedding
$\varphi$
of
$H$
in
$G$
is an injective map
$\varphi \;:\; V(H) \rightarrow V(G)$
such that
$\left \{ v,w \right \}\in E(H)$
implies
$\left \{ \varphi (v),\varphi (w) \right \} \in E(G)$
. For all integers
$a, b$
with
$a\leq b$
, let
$[a,b]\;:\!=\; \left \{ i\in \mathbb{Z}\;:\;a\leq i \leq b \right \}$
and
$[a]\;:\!=\;\left \{ 1, 2,\dots, a \right \}$
. When we write
$\alpha \ll \beta \ll \gamma$
, we always mean that
$\alpha, \beta, \gamma$
are constants in
$(0, 1)$
, and
$\beta \ll \gamma$
means that there exists
$\beta _{0}=\beta _{0}(\gamma )$
such that the subsequent arguments hold for all
$0\lt \beta \leq \beta _{0}$
. Hierarchies of other lengths are defined analogously. For the sake of clarity of presentation, we will sometimes omit floor and ceiling signs when they are not crucial.
2.2. Graph expansion and trees
We will introduce some graph expansion properties to embed the trees.
Definition 2.1 ([Reference Johannsen, Krivelevich and Samotij18]). Let
$n \in \mathbb{N}$
and
$d\gt 0$
. A graph
$G$
is an
$(n,d)$
-expander if
$\vert G \vert =n$
and
$G$
satisfies the following two conditions.
-
1.
$\vert N_{G} (X)\vert \geq d\vert X \vert$
for all sets
$X \subseteq V(G)$
with
$1\leq \vert X \vert \lt \lceil \frac{n}{2d} \rceil$
. -
2.
$e_{G}(X,Y)\gt 0$
for all disjoint
$X,Y \subseteq V(G)$
with
$\vert X \vert =\vert Y \vert =\lceil \frac{n}{2d} \rceil$
.
In [Reference Krivelevich27], Krivelevich considered trees differently according to whether they contain many leaves or many disjoint bare paths.
Lemma 2.2 ([Reference Krivelevich27]). For any integers
$n,k\gt 2$
, a tree on
$n$
vertices either has at least
$\frac{n}{4k}$
leaves or a collection of at least
$\frac{n}{4k}$
vertex-disjoint bare paths of length
$k$
.
We will use the following corollary to divide
$\mathcal{T}(n,\Delta )$
into trees with many pendant stars and trees with many vertex-disjoint caterpillars.
Corollary 2.3.
For any integer
$n,k\gt 2$
, a tree on
$n$
vertices with maximum degree
$\Delta$
either has at least
$\frac{n}{4k\Delta }$
pendant stars or a collection of at least
$\frac{n}{4k\Delta }$
vertex-disjoint caterpillars each of length
$k$
.
Suppose
$T$
is an
$n$
-vertex tree with maximum degree at most
$\Delta$
and
$T^{\prime }$
is the subtree obtained by removing all leaves. Then it holds that
$\vert T^{\prime } \vert +\vert T^{\prime } \vert (\Delta -1)\geq n$
. We can apply Lemma 2.2 on
$T^{\prime }$
to obtain at least
$\frac{|T^\prime |}{4k}\ge \frac{n}{4k\Delta }$
pendant stars or a collection of at least
$\frac{n}{4k\Delta }$
vertex-disjoint caterpillars of length
$k$
in
$T$
.
In [Reference Haxell16], Haxell extended a result of Friedman and Pippenger [Reference Friedman and Pippenger12] and showed that one can embed every almost spanning tree with bounded maximum degree in a graph with strong expansion property. We will use the following result by Johannsen et al. [Reference Johannsen, Krivelevich and Samotij18].
Theorem 2.4 ([Reference Johannsen, Krivelevich and Samotij18]). Let
$n,\Delta \in \mathbb{N}$
,
$d \in \mathbb{R}^+$
with
$d \geq 2\Delta$
and
$G$
be an
$(n,d)$
-expander. Given any
$T \in \mathcal{T}(n-4\Delta \lceil \frac{n}{2d} \rceil, \Delta )$
, we can find a copy of
$T$
in
$G$
.
As shown by Johannsen et al. [Reference Johannsen, Krivelevich and Samotij18], the following lemma is useful for attaching leaves onto certain vertices.
Definition 2.5 ([Reference Johannsen, Krivelevich and Samotij18]). Given a bipartite graph
$G=(A,B,E)$
with
$\vert A\vert \leq \vert B\vert$
and a function
$f\;:\; A\rightarrow \mathbb{N}$
with
$\sum _{u\in A}f(u) = |B|$
, an
$f$
-matching from
$A$
into
$B$
is a collection of vertex-disjoint stars
$\{S_u\;:\;u\in A\}$
in
$G$
such that
$S_u$
has
$u$
as the centre and exactly
$f(u)$
leaves inside
$B$
.
Lemma 2.6 ([Reference Johannsen, Krivelevich and Samotij18]). Let
$d,m\in \mathbb{N}$
and let
$G$
be a graph. Suppose that two disjoint sets
$U,W \subseteq V(G)$
satisfy the following three conditions:
-
1.
$\vert N_{G} (X) \cap W\vert \geq d\vert X \vert$
for all sets
$X \subseteq U$
with
$1\leq \vert X \vert \leq m$
, -
2.
$e_{G}(X,Y)\gt 0$
for all
$X \subseteq U$
and
$Y\subseteq W$
with
$\vert X \vert =\vert Y \vert \geq m$
, -
3.
$d_U(w) \geq m$
for all
$w \in W$
.
Then for every
$f\;:\; U\to \{1,\dots,d\}$
with
$\sum _{u\in U}f(u)=\vert W\vert$
, there exists an
$f$
-matching from
$U$
into
$W$
.
2.3. Proof overview
We give a brief outline of our proof here. Similar to previous works [Reference Krivelevich27, Reference Krivelevich, Kwan and Sudakov28, Reference Montgomery34], we classify the trees and deal with them separately. Our classification is given in Corollary 2.3, which refines Lemma 2.2. First, for the pendant star case, let
$T_1$
be a subtree obtained by deleting the centres and leaves of pendant stars in
$T$
(but not the roots). We randomly partition
$V(G)$
into
$V_{1}, V_{2}$
and
$V_{3}$
and embed
$T_1$
into
$G[V_{1}]$
by Theorem 2.4. Then we greedily embed most of the centres into
$V_{2}$
. The
$\alpha ^*$
property guarantees that we are left with a small portion of centres and we shall embed them by the degree condition into
$V_{3}$
. Finally we use Lemma 2.6 to find a desired star-matching and complete the embedding.
For the caterpillar case, we shall embed a suitable subset of branch vertices into a random set with “good” expansion properties. In this way, we can greedily finish the last step of embedding leaves of caterpillars by a star-matching. To embed this suitable branch vertex set, we control the length of the caterpillars and let
$T_2$
be the subforest obtained by deleting these caterpillars except the ends. First, we randomly partition
$V(G)$
into
$V_1,V_2$
, and
$V_3$
, and embed
$T_2$
into
$V_1$
as the above case. To embed central paths of these caterpillars, we randomly partition
$V_2$
set into
$k-1$
equal parts
$X_1,\dots,X_{k-1}$
, where
$k$
is the length of the caterpillars. Then we define an auxiliary graph
$H$
with
$V(H)=X'_{1}\cup X'_2\cup \cdots \cup X'_{k-2}$
which is a spanning subgraph of the blow-up of
$C_{k-2}$
and
$X'_i$
is roughly the same as
$X_i$
. In this way, we transform the problem of embedding vertex-disjoint central paths into finding a transversal cycle-factor in
$H$
, as Lemma 2.7 below. Once central paths have been embedded, we can greedily embed leaves of caterpillars by a star-matching as mentioned above and complete the embedding. Now we state Lemma 2.7. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_k$
and let
$\alpha ^*_{\textrm{b}}(G)$
be the largest integer
$s$
such that
$G=(V_{1},\ldots,V_{k},E)$
contains an
$(s,s)$
-bipartite-hole
$(S,T)$
where
$S\subseteq V_i, T\subseteq V_{i+1}$
for some
$i\in [k]$
.
Lemma 2.7.
Given a positive integer
$k \in 4\mathbb{N}$
and a constant
$\delta$
with
$\delta \gt \frac{2}{k}$
, there exists
$\alpha \gt 0$
such that the following holds for sufficiently large
$n \in \mathbb{N}$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the n-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq \delta n$
and
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
. Then
$G$
has a transversal
$C_{k}$
-factor.
Although the minimum degree bound in Lemma 2.7 is not best possible and it only works for
$k \in 4\mathbb{N}$
, but it is enough for our purpose (indeed, we only need it work for large
$k$
). We suspect that the (asymptotic) tight condition should be
$(1+o(1))\frac nk$
, given by the so-called space barrier. Indeed, let
$G$
be a (complete)
$n$
-blow-up of
$C_k$
with parts labelled as
$V_{1},\ldots,V_{k}$
. One can specify a set
$U_i$
of size
$n/k-1$
in each cluster
$V_i$
,
$i\in [k]$
and remove all edges not touching
$U\;:\!=\;\bigcup _{i\in [k]} U_i$
from
$G$
. Then we add a
$k$
-partite Erdős graph (obtained from random graph) to
$V(G)\setminus U$
so that the resulting graph
$G'$
satisfy
$\alpha ^*_{\textrm{b}}(G')=o( n)$
but
$G^\prime -U$
is
$C_k$
-free. Now every transversal copy of
$C_k$
in
$G'$
must contain a vertex in
$U$
and
$\bar{\delta }(G')\geq n/k-1$
. Since
$|U|\lt n$
,
$G'$
does not have a transversal
$C_{k}$
-factor.
If we remove the
$\alpha ^*_{\textrm{b}}$
condition in Lemma 2.7, then the minimum degree threshold for transversal
$C_{k}$
-factor is asymptotically
$\bar{\delta }(G)\geq (1+\frac 1k)\frac n2+o(n)$
, as determined recently by Ergemlidze and Molla in [Reference Ergemlidze and Molla9].
Our proof of Lemma 2.7 is based on the absorption method, which will be given in Section 4.4. Finally, there are some other minimum degree-type results in blown-up graphs [Reference Ergemlidze and Molla9, Reference Johansson19] and in multi-partite graphs [Reference Fischer10, Reference Keevash and Mycroft20, Reference Lo and Markström29, Reference Magyar and Martin31, Reference Martin and Szemerédi32], but not with any randomness condition.
3. Proof of the main theorem
Proof of Theorem
1.2. Given a positive integer
$\Delta$
and a constant
$\varepsilon \gt 0$
, we set
$k=48\lceil \frac{1}{\varepsilon } \rceil$
,
$\gamma =\frac{1}{4k\Delta ^2}$
and
$\eta =\frac{1}{4k\Delta }$
. Choose
\begin{equation*} \frac{1}{n}\ll \alpha \ll \frac{1}{d}\ll \varepsilon,\frac{1}{\Delta } \end{equation*}
and let
$G$
be an
$n$
-vertex graph with
$\delta (G)\ge \varepsilon n$
and
$\alpha ^*(G)\lt \alpha n$
. Given any tree
$T\in \mathcal{T}(n,\Delta )$
, by Corollary 2.3, we proceed the proof by considering the following two cases.
Case 1.
$T$
has at least
$\eta n$
pendant stars.
We can easily pick a collection of
$\gamma n$
vertex-disjoint pendant stars in
$T$
and label it as
$\mathcal{D}=\{D_1,\dots,D_{\gamma n}\}$
for convenience. Write
$A\;:\!=\;\left \{ a_{1},\dots,a_{\gamma n} \right \}$
and
$B\;:\!=\;\left \{ b_{1},\dots,b_{\gamma n} \right \}$
where
$a_{i}$
and
$b_{i}$
are the root and centre of
$D_i$
, respectively. Let
$\mathcal{S}=\left \{ S_{1},\dots,S_{\gamma n} \right \}$
where each
$S_{i}$
is obtained from
$D_i$
by removing the root. Let
$T_{1}$
be a subtree of
$T$
obtained by deleting vertices of
$\mathcal{S}$
. Note that
$\vert T_{1}\vert \leq n-2\gamma n$
. Moreover we claim that
$V(G)$
can be partitioned into
$V_{1},V_{2},V_{3}$
of sizes
$n_{1},n_{2},n_{3}$
, respectively, such that
\begin{equation} d_{V_{i}}(v)\geq \frac{\varepsilon }{2}\vert V_{i} \vert{\rm \ for\ all}\ v \in V(G),\ i\in [3] \end{equation}
where
\begin{equation} n_{1}= \frac{d\vert T_{1}\vert +4\Delta d}{d-2\Delta },\quad n_{2}=\gamma n,\quad n_{3}=n-n_{1}-n_{2} . \end{equation}
Choose a partition
$\left \{ V_{1},V_{2},V_{3} \right \}$
of
$V(G)$
uniformly at random where
$\vert V_{i} \vert =n_{i}$
. For every
$v\in V(G)$
, let
$f^{i}_{v}=d_{V_i}(v)$
and note that
$\mu _{v}^{i}\;:\!=\;\mathbb{E}[f_{v}^{i}]\geq \varepsilon n_{i}$
. Let
$q$
be the probability that there exist
$v\in V(G)$
and
$i\in [3]$
which violates property (1). Then by the union bound and Chernoff’s inequality (see e.g. [Reference Janson, Łuczak and Ruciński17], Theorem 2.1),
\begin{equation*} q\leq 3n\exp \left (\frac{-(\mu _{v}^{i}/2)^2}{2\mu _{v}^{i}} \right ) \leq 3n \exp \left (-\frac{\varepsilon n_{i}}{8} \right )=o(1) \end{equation*}
for sufficiently large
$n$
. Therefore, with positive probability the randomly chosen partition
$\left \{ V_{1},V_{2},V_{3} \right \}$
satisfies property (1). Then we have
\begin{align} n_{3}&\geq n-\gamma n-\frac{d(n-2\gamma n)+4\Delta d}{d-2\Delta }=\frac{(d+2\Delta )\gamma n-2\Delta n-4\Delta d}{d-2\Delta } \nonumber \\ &\geq \frac{d\gamma n-2\Delta n}{2d}\geq \frac{\gamma n}{4} \geq \frac{6\Delta \alpha n}{\varepsilon }, \end{align}
where in the penultimate inequality we use
$d\gamma n-2\Delta n\geq \frac{d\gamma n}{2}$
because
$\frac{1}{d}\ll \varepsilon,\frac{1}{\Delta }$
and the last inequality follows since
$\alpha \ll \varepsilon,\frac{1}{\Delta }$
.
Claim 3.1.
$G[V_{1}]$
is an
$(n_1,d)$
-expander.
Proof. Let
$m_{1}=\lceil \frac{\vert V_{1}\vert }{2d}\rceil$
and
$m_{2}=\lceil \frac{\varepsilon \vert V_{1}\vert }{2d+2} \rceil$
. Since
$\alpha ^*(G)\lt \alpha n \leq m_{1}$
, there is at least one edge between any two vertex-disjoint sets of size
$m_1$
in
$G[V_{1}]$
. For
$X \subseteq V_{1}$
with
$1\le \vert X\vert \le m_{2}$
, by (1), we have
\begin{equation} \vert N_{V_{1}}(X)\vert \ge \frac{\varepsilon }{2} \vert V_{1}\vert -\vert X \vert \ge d\vert X\vert . \end{equation}
For
$X\subseteq V_{1}$
with
$m_{2}\lt \vert X\vert \lt m_{1}$
, since
$\alpha \ll \frac{1}{d}$
,
$ \varepsilon$
, we can arbitrarily pick
$Z\subseteq X$
with
$\vert Z\vert =\alpha n$
. As there is no edge between
$Z$
and
$V_{1}\backslash (Z\cup N_{V_{1}}(Z))$
, we have
$\vert V_{1}\backslash (Z\cup N_{V_{1}}(Z))\vert \lt \alpha n$
, then
$\vert N_{V_{1}}(Z)\vert \gt \vert V_{1}\vert -2\alpha n$
. Thus
\begin{equation} \vert N_{V_{1}}(X)\vert \ge \vert N_{V_{1}}(Z)\vert -(\vert X\vert -\vert Z\vert ) \geq \vert V_{1}\vert -\vert X\vert -\alpha n \ge d\vert X\vert . \end{equation}
Together with (4),
$G[V_{1}]$
is an
$(n_1,d)$
-expander.
Note that
$\vert T_{1}\vert = n_{1}-4\Delta (\frac{n_{1}}{2d}+1) \leq n_{1}-4\Delta \lceil \frac{n_{1}}{2d} \rceil$
, where the first equality follows since
$n_{1}= \frac{d\vert T_{1}\vert +4\Delta d}{d-2\Delta }$
. Then by Theorem 2.4, there exists an embedding
$f_{1}\;:\;V(T_1)\to V_1$
. Let
$L_{0}=V_{1}\backslash f_{1}(T_{1})$
and
$L_{1}= f_{1}(A)$
. Next, we will embed the centres of pendant stars into
$V_{2} \cup V_{3}$
and we do it in two steps.
In the first step, we embed most of the vertices of
$B$
into
$V_{2}$
. Consider the bipartite graph
$H$
on vertex sets
$L_{1}$
and
$V_{2}$
, where
$\vert L_{1} \vert =\vert V_{2}\vert =\gamma n$
. Let
$M$
be a maximum matching in
$H$
and we claim that
$\vert E(M)\vert \geq \gamma n-\alpha n$
. Otherwise, since
$\alpha ^*(G)\lt \alpha n$
, there is at least one edge in
$H-V(M)$
, contrary to the maximality of
$M$
. Let
$L_2=V(M)\cap V_2$
and
$L_{3}=V_{2}\setminus L_{2}$
. Without loss of generality, suppose we have embedded
$B_{1}=\left \{ b_{1},\dots,b_{t}\right \}$
into
$L_2$
, where
$t\geq \gamma n-\alpha n$
.
In the second step, we shall embed the vertices of
$B\setminus B_{1}$
into
$V_{3}$
. For every
$u\in V(G)$
, by (1) and (3), we have
$d_{V_{3}}(u)\geq \frac{\varepsilon }{2}\vert V_{3} \vert \geq \Delta \alpha n$
. Hence we can greedily embed
$S_{t+1},\dots,S_{ \gamma n}$
into
$G[V_{3}]$
. Let
$\mathcal{S}_1=\{S_1,\dots,S_t\}$
. It follows that there exists an embedding
$f_{2}\;:\; V(B_{1})\cup V(\mathcal{S}\setminus \mathcal{S}_{1})\rightarrow V_{2}\cup V_{3}$
such that
$f_{2}(B_{1})=L_{2}$
and
$f_{2}(\mathcal{S}\setminus \mathcal{S}_{1})\subseteq V_{3}$
. Let
$L_{4}=V_{3}\setminus f_{2}(\mathcal{S}\setminus \mathcal{S}_{1})$
and we have
$\vert L_{4} \vert \geq \vert V_{3}\vert -\Delta \alpha n$
.
Now it remains to embed the leaves attached to the vertices of
$ B_{1}$
. Consider the bipartite graph
$Q$
on vertex sets
$L_{2}$
and
$L$
, where
$L=L_{0}\cup L_{3}\cup L_{4}$
. Let
$m\;:\!=\;2\alpha n$
and
$d\;:\!=\;\Delta -1$
. Since
$\alpha ^*(G)\lt \alpha n\lt m$
, there is at least one edge between any two disjoint vertex set of size
$m$
in
$Q$
. For all
$X\subseteq L_{2}$
with
$1\leq \vert X\vert \leq m$
, by (1) and (3), we have
$\vert N_{Q}(X)\cap L\vert \geq \vert N_{Q}(X)\cap L_{4}\vert \geq \frac{\varepsilon }{2}\vert V_{3} \vert -\Delta \alpha n\geq d\vert X \vert$
. Moreover for each
$u \in L$
, we have
$d_{ L_{2}}(u)\geq \frac{\varepsilon }{2}\vert V_{2} \vert -\alpha n\geq m$
due to
$\alpha \ll \varepsilon,\frac{1}{\Delta }$
. Therefore by applying Lemma 2.6 on
$Q$
, we obtain an embedding
$f_{3}$
of
$\mathcal{S}_{1}$
in
$Q$
such that
$f_{3}(u)=f_{2}(u)$
for every
$u\in B_{1}$
. In conclusion, it is clear that the map
$f\;:\; V(T)\rightarrow V(G)$
defined by
\begin{equation*} f(u)\;:\!=\;\left \{ \begin{array}{l@{\quad}l} f_{1}(u)& \textrm{if}\; u\in V(T_{1}) \\ f_{2}(u) &\textrm{if}\; u\in V(\mathcal{S})\setminus V(\mathcal{S}_{1}) \\ f_{3}(u) &\textrm{if}\; u\in V(\mathcal{S}_{1}) \\ \end{array} \right . \end{equation*}
is an embedding of
$T$
in
$G$
. The proof of Case 1 is complete.
Case 2.
$T$
has at least
$\eta n$
vertex-disjoint caterpillars of length
$k$
.
A caterpillar in
$T$
consists of a bare path in
$T^{\prime }$
as the central path with some (possibly empty) leaves attached to the internal vertices of the central path, where
$T^\prime$
is the subtree obtained by deleting the leaves of
$T$
and we say that the internal vertices attached with leaves are branch vertices. Observe that
$T$
either has a family of at least
$\frac{\eta n}{2}$
caterpillars of length
$k$
that have at least one leaf or a family of at least
$\frac{\eta n}{2}$
bare paths of length
$k$
. Here, we will give a detailed proof for the first subcase and the second subcase can be derived by the same argument.
Let
$n^{\prime }=\frac{\eta n}{2}$
and
$k^{\prime }$
be an integer from
$\{ \frac{k}{2}, \frac{k}{2}-1, \frac{k}{2}-2, \frac{k}{2}-3\}$
such that
$k^{\prime }=2$
(mod 4). It is easy to pick a collection of
$n^{\prime }$
vertex-disjoint caterpillars of length
$k^{\prime }$
in
$T$
such that one end of each caterpillar is adjacent to a branch vertex in
$T$
. Let
$\mathcal{F}=\left \{ F_{1},\dots,F_{n^{\prime }} \right \}$
be such a collection and
$\mathcal{P}=\left \{ P_{1},\dots,P_{n^{\prime }} \right \}$
, where each
$P_{i}$
is the central path of
$F_{i}$
. Write
$S\;:\!=\;\left \{ s_{1},\dots,s_{n^{\prime }}\right \}$
and
$W\;:\!=\;\left \{ w_{1},\dots,w_{n^{\prime }}\right \}$
where
$s_{i}$
and
$w_{i}$
are the ends of
$F_{i}$
and assume that the neighbour of
$s_i$
in
$P_i$
is a branch vertex,
$i\in [n^{\prime }]$
.
Let
$T_{2}$
be a subforest of
$T$
obtained by deleting the vertices of
$\mathcal{F}$
except the ends of every caterpillar and note that
$ \vert T_2\vert \leq n-n^\prime k^\prime$
. In a similar way as Case 1, there exists a partition
$\left \{ V_{1},V_{2},V_{3} \right \}$
of
$V(G)$
such that
\begin{equation} d_{V_{i}}(u)\geq \frac{\varepsilon }{2}\vert V_{i} \vert{\rm \ for\ all}\ u \in V(G),\ i\in [3] \end{equation}
where
\begin{equation} \vert V_{1}\vert =\vert T_{2} \vert + \frac{2\Delta \vert T_{2}\vert +4\Delta d}{d-2\Delta },\, \vert V_{2}\vert =n^{\prime }(k^{\prime }-1)-\frac{2\Delta \vert T_{2}\vert +4\Delta d}{d-2\Delta }, \, \vert V_{3}\vert = n-\vert V_{1}\vert -\vert V_{2}\vert . \end{equation}
Then we have
\begin{equation} \vert V_{3}\vert \geq n-(n-n^{\prime }k^{\prime })-n^{\prime }(k^{\prime }-1) = n^{\prime } \geq \frac{2\Delta \alpha n}{\varepsilon }, \end{equation}
where the first inequality follows since
$\vert T_2\vert \leq n-n^\prime k^\prime$
and the last inequality follows since
$\alpha \ll \varepsilon,\frac{1}{\Delta }$
.
Note that
$\vert T_{2} \vert = \vert V_{1}\vert -4\Delta (\frac{\vert V_{1}\vert }{2d}+1)\leq \vert V_{1}\vert -4\Delta \lceil \frac{\vert V_{1}\vert }{2d}\rceil$
, where the first equality follows since
$\vert V_{1}\vert =\vert T_{2} \vert + \frac{2\Delta \vert T_{2}\vert +4\Delta d}{d-2\Delta }$
. Then Theorem 2.4 implies that there exists an embedding
$g_1$
of
$T_{2}$
in
$G[V_{1}]$
. Let
$L_{1}=V_{1}\backslash g_{1}(T_{2})$
and
$V_{2}^{\prime }=V_2\cup L_1$
. It now remains to embed
$n^{\prime }$
caterpillars in
$\mathcal{F}$
.
First, we shall embed the internal vertices of
$\mathcal{P}$
into
$V_{2}^{\prime }$
. Randomly partition
$V_2^{\prime }$
into
$X_1,\dots,X_{k^{\prime }-1}$
each of size
$n^{\prime }$
. The union bound and Chernoff’s inequality imply that, if
$n$
is sufficiently large, then there exists a partition such that for every
$u\in V(G)$
and
$i\in [k^\prime -1]$
, we have
$d_{X_i}(u)\ge \frac{\varepsilon n^{\prime }}{4}$
. Since
$\alpha ^*(G)\lt \alpha n\leq n^{\prime }$
, there exists a matching
$M_{1}$
between
$g_{1}(S)$
and
$X_{1}$
such that
$t_1\;:\!=\;\vert E(M_{1})\vert \gt n^{\prime }-\alpha n$
. Similarly, there exists a matching
$M_{2}$
between
$g_1(W)$
and
$X_{k^{\prime }-1}$
such that
$t_2\;:\!=\;\vert E(M_{2})\vert \gt n^{\prime }-\alpha n$
. Let
$S_1=\{s_1,\dots,s_{t_1}\}\subseteq S$
and
$W_1=\{w_1,\dots,w_{t_2}\}\subseteq W$
. Without loss of generality, we assume that
$g_1(S_1)\subseteq V(M_{1})$
and
$g_1(W_1)\subseteq V(M_{2})$
. By the choice of
$X_1,\dots, X_{k^\prime -1}$
, for each
$u\in g_1((S\cup W)G^\prime -Us (S_{1}\cup W_1))\subseteq V(G)$
, we have
\begin{equation*} d_{X_2}(u) \geq \frac{\varepsilon n^{\prime }}{4} \ge 2\alpha n \geq (n^{\prime }-t_1)+(n^\prime -t_2). \end{equation*}
Therefore we can greedily find a matching
$M_3$
between
$g_1(S\setminus S_{1})$
and
$X_2$
covering
$g_1(S\setminus S_{1})$
, and a matching
$M_4$
between
$g_1(W\setminus W_{1})$
and
$X_2$
covering
$g_1(W\setminus W_{1})$
, where
$V(M_3)\cap V(M_4)=\emptyset$
. Let
$X_1^\prime \;:\!=\;(V(M_1)\cap X_1)\cup (V(M_3)\cap X_2)$
,
$X_{k^\prime -1}^\prime \;:\!=\;(V(M_2)\cap X_{k^\prime -1})\cup (V(M_4)\cap X_2)$
,
$X_2^\prime \;:\!=\;(X_2 \setminus (V(M_3)\cup V(M_4))) \cup (X_1\setminus V(M_1))\cup (X_{k^\prime -1}\setminus V(M_2))$
and let
$X_i^\prime \;:\!=\;X_i$
for
$i\in [3,k^\prime -2]$
. In this way, we obtain a new partition
$\left \{ X_{1}^\prime,\dots,X^\prime _{k^\prime -1} \right \}$
of
$V_{2}^{\prime }$
such that there exist perfect matchings between
$X_1^\prime$
and
$g_1(S)$
and between
$X^\prime _{k^{\prime }-1}$
and
$g_1(W)$
. Let
$X_{1}^\prime =\left \{ x_{1},\dots,x_{n^{\prime }}\right \}$
and
$X^\prime _{k^{\prime }-1}=\left \{ y_{1},\dots,y_{n^{\prime }}\right \}$
where for each
$i\in [n^{\prime }]$
,
$\{x_{i},g_1(s_{i})\}$
and
$\{y_{i},g_1(w_{i})\}$
are edges of the above perfect matchings.
Let
$X^\prime _{0}=\left \{ z_{1},\dots,z_{n^{\prime }}\right \}$
be a new set of vertices disjoint from
$V(G)$
. Define an auxiliary graph
$H$
with vertex set
$V(H)=X^\prime _{0}\cup X^\prime _{2}\cup \cdots \cup X^\prime _{k^{\prime }-2}$
. For
$2\leq i\leq k^{\prime }-3$
, the edges of
$H$
between
$X^\prime _{i}$
and
$X^\prime _{i+1}$
are identical to those of
$G$
. For
$v\in X^\prime _{2}$
and
$z_{j}\in X^\prime _{0}$
,
$\left \{ v,z_{j}\right \}$
is an edge of
$H$
if and only if
$\left \{v,x_{j}\right \}$
is an edge of
$G$
. Similarly, for
$u\in X^\prime _{k^{\prime }-2}$
and
$z_{j}\in X^\prime _{0}$
,
$\left \{ u,z_{j}\right \}$
is an edge of
$H$
if and only if
$\left \{ u,y_{j}\right \}$
is an edge of
$G$
. Observe that if
$H$
has a transversal
$C_{k^{\prime }-2}$
-factor, then
$G$
has
$n^{\prime }$
vertex-disjoint paths of length
$k^{\prime }-2$
that connect
$x_{i}$
and
$y_{i}$
. Since we moved at most
$2\alpha n$
vertices when we constructed the new partition, now we have that
\begin{equation} \bar{\delta }(H) \geq \frac{\varepsilon n^{\prime }}{4}-2\alpha n \geq \frac{\varepsilon n^{\prime }}{8} \gt \frac{2n^\prime }{k^\prime -2}. \end{equation}
Then Lemma 2.7 implies that
$H$
contains a transversal
$C_{k^{\prime }-2}$
-factor. Together with the perfect matchings, we find an embedding
$g_2$
of
$V(\mathcal{P})$
to
$V_2^{\prime }$
that connects
$g_1(s_{i})$
and
$g_1(w_{i})$
for each
$i\in [n^{\prime }]$
. Now it suffices to embed the leaves of caterpillars into
$V_{3}$
.
Let
$I\subseteq V_2^{\prime }$
be the set of images of branch vertices in
$V(\mathcal{F})$
. Since every vertex in
$S$
is adjacent to a branch vertex in
$F_i$
, we have
$X_{1}^\prime \subseteq I$
. Let
$m\;:\!=\;\alpha n$
and
$d\;:\!=\;\Delta$
. For all
$X\subseteq I$
with
$1\leq \vert X\vert \leq m$
, by (6) and (8), we have
$\vert N_{G}(X)\cap V_{3}\vert \geq \frac{\varepsilon }{2}\vert V_{3} \vert \geq d\vert X\vert$
. Moreover for each
$u \in V_{3}$
, by (9), we have
$d_{I}(u) \geq d_{X^\prime _{1}}(u)\geq \frac{\varepsilon n^{\prime } }{8} \geq m$
. Together with the assumption that
$\alpha ^*(G)\lt m$
, Lemma 2.6 implies that there exists an embedding
$g_{3}$
of
$V(\mathcal{F})\backslash V(\mathcal{P})$
to
$V_{3}$
that respects the edges between the branch vertices and the leaves of caterpillars. It follows that the map
$g\;:\; V(T)\rightarrow V(G)$
defined by
\begin{equation*} g(u)\;:\!=\;\left \{ \begin{array}{l@{\quad}l} g_{1}(u)& \textrm{if}\; u\in V(T_{2}) \\ g_{2}(u) &\textrm{if}\; u\in V(\mathcal{P}) \\ g_{3}(u) &\textrm{if}\; u\in V(\mathcal{F})\backslash V(\mathcal{P}) \\ \end{array} \right . \end{equation*}
is an embedding of
$T$
in
$G$
. This concludes the proof of the first subcase.
As for the second subcase, we adopt a similar argument and the main difference is that we split
$V(G)$
into two parts because the caterpillars have no leaves and it suffices to find
$g_1$
and
$g_2$
.
4. Transversal
$C_{k}$
-factor
4.1. Proof of Lemma 2.7
Following the typical absorption method, the main tasks are to (
$i$
) establish an absorbing set
$R$
and (
$ii$
) find an almost perfect transversal
$C_{k}$
-tiling in
$G-R$
. For (
$i$
), we will introduce some related definitions.
Definition 4.1. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_{k}$
and
$F$
be a
$k$
-vertex graph.
-
1. We say that a balanced subset
$R\subseteq V(G)$
is a
$\xi$
-absorbing set for some
$\xi \gt 0$
if for every balanced subset
$U\subseteq V(G)\backslash R$
with
$\vert U\vert \leq \xi n$
,
$G[R\cup U]$
contains an
$F$
-factor which consists of transversal copies. -
2. Given a subset
$S\subseteq V(G)$
of size
$k$
and an integer
$t$
, we say that a subset
$A_{S}\subseteq V(G) \backslash S$
is an
$(F,t)$
-absorber of
$S$
if
$\vert A_{S}\vert \leq kt$
and both
$G[A_{S}]$
and
$G[A_{S}\cup S]$
contain an
$F$
-factor.
Now we state the first crucial lemma, whose proof can be found in Section 4.4.2.
Lemma 4.2. (Absorbing Lemma). Given
$k \in \mathbb{N}$
with
$k\geq 4$
and positive constants
$\delta,\gamma$
with
$\delta \gt \frac{2}{k}$
and
$\gamma \leq \frac{\delta }{2}$
, there exist
$\alpha, \xi \gt 0$
such that the following holds for sufficiently large
$n \in \mathbb{N}$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq \delta n$
and
$\alpha ^*_{\mathrm{b}}(G)\lt \alpha n$
. Then there exists a
$\xi$
-absorbing set
$R\subseteq V(G)$
of size at most
$\gamma n$
.
For (
$ii$
), Lemma 4.3 provides an almost transversal
$C_{k}$
-tiling, whose proof will be given in Section 4.3.
Lemma 4.3. (Almost perfect tiling). Given a positive integer
$k\in 4\mathbb{N}$
and constants
$\delta,\zeta$
with
$\delta \gt \frac{2}{k}$
, there exists
$\alpha \gt 0$
such that the following holds for sufficiently large
$n \in \mathbb{N}$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the n-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq \delta n$
and
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
. Then
$G$
contains a transversal
$C_{k}$
-tiling covering all but at most
$\zeta n$
vertices.
Now we are ready to prove Lemma 2.7 using Lemmas 4.2 and 4.3.
Proof of Lemma
2.7. Given
$k \in 4\mathbb{N}$
and a constant
$\delta$
with
$\delta \gt \frac{2}{k}$
, we set
$\eta \;:\!=\;\delta -\frac{2}{k}$
and choose
$\frac{1}{n}\ll \alpha \ll \zeta \ll \xi \ll \gamma \ll \eta,\delta$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq (\frac{2}{k}+\eta ) n$
and
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
.
By Lemma 4.2 and the choice that
$\gamma \ll \eta,\delta$
, there exists a
$\xi$
-absorbing set
$R\subseteq V(G)$
of size at most
$\gamma n$
for some
$\xi \gt 0$
. Let
$G^{\prime }\;:\!=\;G-R$
and note that
$G^\prime$
is an
$(n-\frac{\vert R\vert }{k})$
-blow-up of
$C_k$
. Then we have
\begin{equation*} \bar{\delta }(G^{\prime })\geq \left (\frac{2}{k}+\eta \right ) n-\frac{\gamma n}{k}\geq \left (\frac{2 }{k}+\frac{\eta }{2} \right ) n. \end{equation*}
Therefore by applying Lemma 4.3 on
$G^{\prime }$
, we obtain a transversal
$C_{k}$
-tiling
$\mathcal{M}$
that covers all but a set
$U$
of at most
$\zeta n$
vertices in
$G^{\prime }$
. Since
$\zeta \ll \xi$
, the absorbing property of
$R$
implies that
$G[R\cup U]$
contains a transversal
$C_{k}$
-factor, which together with
$\mathcal{M}$
forms a transversal
$C_{k}$
-factor in
$G$
.
4.2. Regularity
The proof of Lemma 4.3 is based on a standard application of the regularity method. We will introduce some basic definitions and properties. Given a graph
$G$
and a pair
$(X,Y)$
of vertex-disjoint subsets in
$V(G)$
, the density of
$(X,Y)$
is defined as
\begin{equation*} d(X,Y)=\frac{e(X,Y)}{\vert X \vert \vert Y \vert }. \end{equation*}
Given constants
$\varepsilon,d\gt 0$
, we say that
$(X,Y)$
is
$(\varepsilon,d)$
-regular if
$d(X,Y)\geq d$
and for all
$X^{\prime } \subseteq X$
,
$Y^{\prime } \subseteq Y$
with
$\vert X^{\prime } \vert \geq \varepsilon \vert X \vert$
and
$\vert Y^{\prime } \vert \geq \varepsilon \vert Y \vert$
, we have
\begin{equation*} \vert d(X^{\prime },Y^{\prime })-d(X,Y)\vert \leq \varepsilon . \end{equation*}
The following fact results from the definition.
Fact 4.4. Let
$(X,Y)$
be an
$(\varepsilon,d)$
-regular pair and
$B \subseteq Y$
with
$\vert B\vert \ge \varepsilon \vert Y\vert .$
Then all but at most
$\varepsilon \vert X\vert$
vertices in
$X$
have at least
$(d-\varepsilon )\vert B\vert$
neighbours in
$B$
.
We now state a degree form of the regularity lemma (see [Reference Komlós and Simonovits26], Theorem 1.10).
Lemma 4.5. (Degree form of Regularity Lemma [Reference Komlós and Simonovits26]). For every
$\varepsilon \gt 0$
there is an
$N = N(\varepsilon )$
such that the following holds for any real number
$d\in (0,1]$
and
$n\in \mathbb{N}$
. Let
$G=(V, E)$
be an
$n$
-vertex graph. Then there exists a partition
$\mathcal{P}=V_{0}\cup V_{1}\cup \dots \cup V_{k}$
and a spanning subgraph
$G^{\prime } \subseteq G$
with the following properties:
-
(a)
$\frac{1}{\varepsilon } \leq k \leq N$
; -
(b)
$\vert V_{0} \vert \leq \varepsilon n$
and
$\vert V_{1} \vert =\dots =\vert V_{k} \vert =m\leq \varepsilon n$
; -
(c)
$d_{G^{\prime }}(v)\geq d_{G}(v)-(d+\varepsilon )n$
for all
$v \in V(G)$
; -
(d) every
$V_{i}$
is an independent set in
$G^{\prime }$
for
$i\in [k]$
; -
(e) every pair
$(V_{i}, V_{j})$
,
$1 \leq i\lt j \leq k$
is
$\varepsilon$
-regular in
$G^{\prime }$
with density
$0$
or at least
$d$
.
A widely used auxiliary graph accompanied with the regular partition is the reduced graph. The reduced graph
$R_{d}$
of
$\mathcal{P}$
is a graph defined on the vertex set
$\left \{ V_{1},\ldots,V_{k} \right \}$
such that
$V_{i}$
is adjacent to
$V_{j}$
in
$R_{d}$
if
$(V_{i},V_{j})$
has density at least
$d$
in
$G^{\prime }$
. We use
$d_{R}(V_{i})$
to denote the degree of
$V_{i}$
in
$R_{d}$
for each
$i\in [k]$
.
Fact 4.6. Given positive constants
$d,\varepsilon$
and
$\delta$
, fix an
$n$
-vertex graph
$G=(V,E)$
with
$\delta (G)\geq \delta n$
and let
$G^{\prime }$
and
$\mathcal{P}$
be obtained by Lemma 4.5, and
$R_{d}$
be given as above. Then for every
$V_{i}\in V(R_{d})$
, we have
$d_{R_d}(V_{i})\geq (\delta -2\varepsilon -d)k$
.
4.3. Almost perfect tilings
Here we shall make use of the following result which provides a sufficient condition for a transversal path among given sets.
Proposition 4.7.
Given an integer
$k\ge 2$
and a positive constant
$\alpha \leq \frac{1}{2}$
, let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_k$
with
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
. For any integers
$i,j$
with
$1\leq i\lt j\leq k$
and a collection of subsets
$X_{s}\subseteq V_s$
with
$s\in [i,j]$
, if
$\vert X_{i}\vert$
,
$\vert X_{j}\vert \geq \alpha n$
and
$\vert X_{\ell }\vert \geq 2\alpha n$
for
$\ell \in [i+1,j-1]$
(possibly empty), then there exists a transversal path
$x_{i}x_{i+1}\dots x_{j-1}x_{j}$
where
$x_{s}\in X_{s}$
for
$s\in [i,j]$
.
Proof of Proposition
4.7. Without loss of generality, we may take
$i=1$
,
$j=k$
for instance. Let
$Z_{1}\;:\!=\;X_{1}$
and
$Z_{2}\;:\!=\;N(Z_{1})\cap X_{2}$
. By the fact that
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n\leq \vert Z_{1} \vert, \vert X_{2} \vert$
, it holds that
$\vert Z_{2}\vert \gt \vert X_{2}\vert -\alpha n\geq \alpha n$
. If
$k=2$
, then there exists an edge
$\left \{ x_{1},x_{2} \right \}$
between
$Z_1$
and
$Z_2$
and we are done. If
$k\gt 2$
, then for each
$s\in [k-1]$
, there exist
$Z_{s}\subseteq X_{s}$
of size larger than
$\alpha n$
such that
$Z_{s}$
is the set of neighbours of
$Z_{s-1}$
. Since
$\vert Z_{k-1}\vert$
,
$\vert X_{k} \vert \geq \alpha n$
, there exists an edge
$\left \{ x_{k-1},x_{k} \right \}$
between
$Z_{k-1}$
and
$ X_{k}$
. Therefore, we can find a transversal path
$x_{1}x_{2}\dots x_{k-1}x_{k}$
, where
$x_{s}\in X_{s}$
for
$s\in [k]$
.
Now we are ready to prove Lemma 4.3.
Proof of Lemma
4.3. Given
$k\in 4\mathbb{N}$
and
$\delta,\zeta$
with
$\delta \gt \frac{2}{k}$
, we set
$\eta \;:\!=\;\delta -\frac{2}{k}$
and choose
$\frac{1}{n}\ll \alpha \ll \frac{1}{N_{0}}\ll \varepsilon \ll \zeta, \delta,\eta$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq \delta n$
and
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
. By applying Lemma 4.5 on
$G$
with
$d\;:\!=\;\frac{\eta }{4}$
, we obtain a partition
$\mathcal{P}=\left \{ U_{0} \right \}\cup \left \{ U_{i,j}\subseteq V_{i}\;:\; i\in [k], j\in [N_{0}] \right \}$
that refines the partition
$\{V_1,V_2,\dots,V_k\}$
of
$G$
and a spanning subgraph
$G^{\prime }$
of
$G$
with properties
$(a)-(e)$
, where we write
$m\;:\!=\;\vert U_{i,j} \vert$
for all
$i\in [k]$
,
$j\in [N_{0}]$
. Let
$R_{d}$
be the reduced graph defined on the vertex set
$\left \{ U_{i,j}\;:\; i\in [k], j\in [N_{0}] \right \}$
. For each
$i\in [k]$
, let
$\mathcal{V}_{i}=\left \{ U_{i,j}\;:\; j\in [N_{0}] \right \}$
and note that
$\{\mathcal{V}_i\;:\;i\in [k]\}$
is a partition of
$R_d$
. Then Fact 4.6 implies that
$\bar{\delta }(R_{d})\geq (\delta -\frac{\eta }{2})N_{0}=(\frac{2}{k}+\frac{\eta }{2})N_{0}$
.
To obtain an almost perfect transversal
$C_{k}$
-tiling in
$G$
, we define an auxiliary graph
$H$
with
$k$
vertices and two disjoint edges and then use it for embedding copies of transversal
$C_k$
(see Claim 4.9). Now we will show that there exist
$N_{0}$
vertex-disjoint copies of specific
$H$
in
$R_{d}$
.
Claim 4.8.
For every
$i\in [k-1]$
,
$R_{d}[\mathcal{V}_{i},\mathcal{V}_{i+1}]$
has a matching of size
$\min \{N_0, 2\bar \delta (R_{d})\}$
.
Proof. Let
$m=\min \{N_0, 2\bar \delta (R_{d})\}$
and without loss of generality, we may take
$i=1$
for instance. Let
$M$
be a maximum matching in
$R_{d}[\mathcal{V}_{1},\mathcal{V}_{2}]$
and assume for the contrary that
$\vert E(M)\vert \leq m-1$
. Let
$U$
be the minimum vertex cover of
$R_{d}[\mathcal{V}_{1},\mathcal{V}_{2}]$
. Then by König’s theorem ([Reference Lovász and Plummer30]), it holds that
$\vert U \vert =\vert E(M) \vert$
. We write
$A=U\cap \mathcal{V}_{1}$
and
$B=U\cap \mathcal{V}_{2}$
. By the pigeonhole principle, we get
$\vert A\vert \leq \lfloor \frac{m-1}{2}\rfloor$
or
$\vert B\vert \leq \lfloor \frac{m-1}{2}\rfloor$
. Suppose
$\vert A\vert \leq \lfloor \frac{m-1}{2}\rfloor$
. Since
$m\leq N_0$
, we have
$\vert \mathcal{V}_{2}\backslash B \vert \neq 0$
. Arbitrarily choose
$u\in \mathcal{V}_{2}\backslash B$
, then
$u$
has no neighbour in
$\mathcal{V}_{1}\backslash A$
. Thus we have
$d_{\mathcal{V}_{1}}(u)=d_{A}(u)\leq \lfloor \frac{m-1}{2}\rfloor \leq \lfloor \frac{2\bar \delta (R_{d})-1}{2}\rfloor = \bar{\delta }(R_{d})-1$
, a contradiction.
By Claim 4.8, for each
$i\in [\frac{k}{2}]$
, there exists a matching
$M_{i}$
of size
$\min \{N_0, 2\bar \delta (R_{d})\}$
between
$\mathcal{V}_{2i-1}$
and
$\mathcal{V}_{2i}$
. Let
$A_{j}\subseteq \mathcal{V}_{j}$
be the vertices uncovered by matchings for
$j\in [k]$
. We pick a family
$\mathcal{H}$
of
$N_{0}$
vertex-disjoint copies of
$H$
such that each copy contains two disjoint edges
$e_{1}$
and
$e_{2}$
where
$e_{1}\in M_{2i-1}$
and
$e_{2}\in M_{2i}$
for some
$i\in [\frac{k}{4}]$
and exactly one vertex inside each
$A_{j}$
for
$j\in [k]\backslash \left \{4i-3,4i-2, 4i-1,4i\right \}$
. Since
\begin{equation*} \sum _{i=1}^{\frac{k}{4}}\vert M_{2i-1} \vert = \frac{k}{4}\cdot \min \{N_0, 2\bar \delta (R_{d})\} \geq N_{0}, \end{equation*}
and similarly
$\sum _{i=1}^{k/4} \vert M_{2i}\vert \ge N_0$
, we can greedily find
$N_{0}$
vertex-disjoint copies of
$H$
that together cover all vertices in
$R_{d}$
.
Claim 4.9.
For each copy of
$H$
in
$\mathcal{H}$
, we can find a transversal
$C_{k}$
-tiling covering all but at most
$\frac{\zeta m}{2}$
vertices in the union of its clusters in
$G$
.
Proof of Claim
4.9. Given a copy of
$H$
, without loss of generality, we may assume that
$V(H)=\left \{U_{1,1}, U_{2,2},\ldots,U_{k,k}\right \}$
and
$\left \{U_{1,1}, U_{2,2}\right \}$
,
$\left \{U_{3,3}, U_{4,4}\right \} \in E(H)$
. Therefore
$(U_{1,1},U_{2,2})$
and
$(U_{3,3},U_{4,4})$
are
$(\varepsilon,d)$
-regular in
$G^\prime$
. Now it suffices to show that for any
$Z_{i}\subseteq U_{i,i}$
with
$i\in [k]$
each of size at least
$\frac{\zeta m}{2k}$
, there exists a copy of
$C_{k}$
with exactly one vertex inside each
$Z_{i}$
.
Since
$\vert Z_{i} \vert \geq \frac{\zeta m}{2k} \ge \varepsilon m$
for
$i\in [4]$
, Fact 4.4 implies that there exists a subset
$Z_{2}^{\prime }\subseteq Z_{2}$
of size at least
$\vert Z_{2} \vert -\varepsilon m$
such that every vertex in
$Z_{2}^{\prime }$
has at least
$(d-\varepsilon )\vert Z_{1} \vert$
neighbours inside
$Z_{1}$
and a subset
$Z_{3}^{\prime }\subseteq Z_{3}$
of size at least
$\vert Z_{3} \vert -\varepsilon m$
such that every vertex in
$Z_{3}^{\prime }$
has at least
$(d-\varepsilon )\vert Z_{4} \vert$
neighbours inside
$Z_{4}$
respectively.
By the assumption
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
and the fact that
$\vert Z_2^\prime \vert, \vert Z_3^\prime \vert \ge \frac{\zeta m}{2k}-\varepsilon m \ge \alpha n$
, there is at least one edge between
$Z_{2}^{\prime }$
and
$Z_{3}^{\prime }$
. Arbitrarily choose one edge
$\left \{u,v\right \}$
with
$u\in Z_2'$
and
$v\in Z_3'$
, and let
$Q_{1}=N(u)\cap Z_{1}$
and
$Q_{2}=N(v)\cap Z_{4}$
, respectively. Then we have
$\vert Q_{1} \vert,\vert Q_{2} \vert \geq (d-\varepsilon )\cdot \frac{\zeta m}{2k}\geq \alpha n$
and note that
$\vert Z_i\vert \ge \frac{\zeta m}{2k}\ge 2\alpha n$
for every
$i\in [k]$
. By applying Proposition 4.7 with
$X_{i}\;:\!=\;Q_{1}$
and
$X_{j}\;:\!=\;Q_{2}$
, we can obtain a transversal path of length
$k-3$
, where the ends in
$Q_{1}$
and
$Q_{2}$
are denoted by
$u^{\prime }$
and
$v^{\prime }$
, respectively. Together with three edges
$\left \{u^{\prime },u\right \}$
,
$\left \{u,v\right \}$
,
$\left \{v,v^{\prime }\right \}$
, we can construct a copy of transversal
$C_{k}$
in
$\cup _{i=1}^{k}Z_{i}$
. Thus we can obtain a transversal
$C_{k}$
-tiling covering all but at most
$\frac{\zeta m}{2}$
vertices in
$\cup _{i=1}^{k} U_{i,i}$
in
$G$
.
This would finish the proof as the union of these
$C_{k}$
-tilings taken over all copies of
$H$
in
$\mathcal{H}$
would leave at most
\begin{equation*} \vert U_{0}\vert +\vert \mathcal{H} \vert \cdot \frac{\zeta m}{2} \leq \varepsilon n+\frac{\zeta n }{2}\leq \zeta n \end{equation*}
vertices uncovered.
4.4. Building an absorbing set
A typical step in the absorption method for
$F$
-factor is to show that every
$k\;:\!=\;\vert V(F) \vert$
-set has polynomially many absorbers (see [Reference Han, Hu, Wang and Yang14]). However, it remains unclear whether this property holds in our setting. Instead, a new approach due to Nenadov and Pehova [Reference Nenadov and Pehova35] guarantees an absorbing set provided that every
$k$
-set has linearly many vertex-disjoint absorbers. Since the host graph in Lemma 4.2 is
$k$
-partite, we aim to show that every transversal
$k$
-set has linearly many vertex-disjoint absorbers. For this, we shall make use of the lattice-based absorbing method developed by Han [Reference Han13].
4.4.1. Finding absorbers
To illustrate the lattice-based absorbing method, we introduce some definitions. Let
$G, F$
be given as aforementioned and
$m, t$
be positive integers. Then we say that two vertices
$u,v\in V(G)$
are
$(F, m, t)$
-reachable (in
$G$
) if for any set
$W$
of
$m$
vertices, there is a set
$S\subseteq V(G)\backslash W$
of size at most
$kt-1$
such that both
$G[\!\left \{u \right \}\cup S]$
and
$G[\!\left \{v \right \}\cup S]$
have
$F$
-factors, where we call such
$S$
an
$F$
-
$\textit{connector}$
for
$u, v$
. Moreover, a set
$U\subseteq V(G)$
is
$(F, m, t)$
-
$\textit{closed}$
if every two vertices
$u, v$
in
$U$
are
$(F, m, t)$
-reachable, where the corresponding
$F$
-connector for
$u, v$
may not be included in
$U$
.
The following result builds a sufficient condition to ensure that every transversal
$k$
-set has linearly many vertex-disjoint absorbers.
Lemma 4.10.
Given
$k \in \mathbb{N}$
with
$k\geq 4$
and a constant
$\delta \gt \frac{2}{k}$
, there exist
$\alpha,\beta \gt 0$
such that the following holds for sufficiently large
$n \in \mathbb{N}$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the n-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq \delta n$
and
$\alpha ^*_{\mathrm{b}}(G)\lt \alpha n$
. Then every transversal
$k$
-set in
$G$
has at least
$ \frac{\beta n-k}{4k^2}$
vertex-disjoint
$(C_{k},2k)$
-absorbers.
4.4.2. Proof of Lemma 4.2
In order to prove the existence of an absorbing set, we introduce a notion of
$F$
-fan.
Definition 4.11. ([Reference Han, Morris, Wang and Yang15]) For a vertex
$v\in V(G)$
and a
$k$
-vertex graph
$F$
, an
$F$
-fan
$\mathcal{F}_{v}$
at
$v$
in
$V(G)$
is a collection of pairwise disjoint sets
$S\subseteq V(G)\backslash \left \{v \right \}$
such that for each
$S\in \mathcal{F}_{v}$
we have that
$\vert S\vert =k-1$
and
$\left \{v \right \}\cup S$
spans a copy of
$F$
.
To build an absorbing structure, we shall make use of bipartite templates as follows, which was introduced by Montgomery [Reference Montgomery34].
Lemma 4.12.
Let
$\beta \gt 0$
. There exists
$m_{0}$
such that the following holds for every
$m\geq m_{0}$
. There exists a bipartite graph
$B_{m}$
with vertex classes
$X_{m}\cup Y_{m}$
and
$Z_{m}$
and maximum degree 40, such that
$\vert X_{m}\vert =m+\beta m$
,
$\vert Y_{m}\vert =2m$
and
$\vert Z_{m}\vert =3m$
, and for every subset
$X^{\prime }_{m}\subseteq X_{m}$
of size
$\vert X^{\prime }_{m}\vert =m$
, the induced graph
$B[X^{\prime }_{m}\cup Y_{m},Z_{m}]$
contains a perfect matching.
Proof of Lemma
4.2. Given
$k \in \mathbb{N}$
with
$k\geq 4$
and positive constants
$\delta \gt \frac{2}{k}$
and
$\gamma \leq \frac{\delta }{2}$
, we choose
$\frac{1}{n}\ll \alpha \ll \xi \ll \beta \ll \delta,\gamma, \frac{1}{k}$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq \delta n$
and
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
. Lemma 4.10 implies that every transversal
$k$
-set in
$G$
has at least
$\frac{\beta n-k}{4k^2}$
vertex-disjoint
$(C_{k},2k)$
-absorbers. Let
$\tau \;:\!=\;\frac{\beta }{8k^2}$
. Then for every
$v\in V(G)$
, there is a
$C_k$
-fan
$\mathcal{F}_{v}$
in
$V(G)$
of size at least
$ \frac{\beta n-k}{4k^2}\ge \tau n$
. Now it suffices to find a
$\xi$
-absorbing set
$R$
for some
$\xi \gt 0$
such that
$\vert R \vert \leq \tau n\leq \gamma n$
.
Let
$q=\frac{\tau }{1000k^3}$
and
$\beta ^{\prime }=\frac{q^{k-1}\tau }{2k}$
. For
$i\in [k]$
, let
$X_{i}\subseteq V_{i}$
be a set of size
$qn$
chosen uniformly at random. For every
$v\in V(G)$
, let
$f_{v}$
denote the number of the sets from
$\mathcal{F}_{v}$
that lie inside
$\cup _{i=1}^{k} X_{i}$
. Note that
$\mu \;:\!=\;\mathbb{E}[f_{v}]= q^{k-1}\vert \mathcal{F}_{v} \vert \geq q^{k-1}\tau n$
. By the union bound and Chernoff’s inequality, we have
\begin{equation*} \mathbb{P}\left [\,\textrm{there}\ \textrm{is}\ v\in V(G)\ \textrm{with}\ f_{v}\lt \frac{\mu }{2}\,\right ]\leq kn\, \textrm{exp}\left (\frac{-(\mu/2)^2}{2\mu }\right )\leq kn\, \textrm{exp}\left (-\frac{q^{k-1}\tau }{8}n\right )=o(1). \end{equation*}
Therefore, as
$n$
is sufficiently large, there exist
$X_{i}\subseteq V_{i}$
with
$\vert X_{i} \vert = qn$
such that for each
$v\in V(G)$
, there is a subfamily
$\mathcal{F}_{v}^{\prime }$
of at least
$\frac{q^{k-1}\tau n}{2}= k\beta ^{\prime }n$
sets from
$\mathcal{F}_{v}$
contained in
$\cup _{i=1}^{k} X_{i}$
.
Let
$m=\vert X_{i} \vert/(1+\beta ^{\prime })$
and note that
$m$
is linear in
$n$
. Let
$\left \{ I_{i} \right \}_{i\in [k]}$
be a partition of
[$3\textit{km}$]
with each
$\vert I_{i} \vert =3{m}$
. For
$i\in [k]$
, arbitrarily choose
$k$
vertex-disjoint subsets
$Y_{i}$
,
$ Z_{i,j}$
for
$j\in [k]\backslash \left \{ i\right \}$
in
$V_{i}\backslash X_{i}$
with
$\vert Y_{i}\vert =2{m}$
and
$\vert Z_{i,j}\vert =3{m}$
. Let
$X=\cup _{i=1}^{k} X_{i}$
,
$Y=\cup _{i=1}^{k} Y_{i}$
and
$ Z=\cup Z_{i,j}$
. Then we have
$\vert X\vert =(1+\beta ^{\prime }){km}$
,
$\vert Y\vert = 2{km}$
and
$\vert Z \vert =3k(k-1)m$
. For each
$j\in [k]$
, we partition
$\cup _{i\in [k]\backslash \left \{ j \right \}}Z_{i,j}$
into a family
$\mathcal{Z}_j$
of
$3m$
transversal
$(k-1)$
-sets and take an arbitrary bijection
$\phi _{j}\;:\;\mathcal{Z}_j\rightarrow I_{j}$
. Moreover, we define a function
$\varphi$
on
$[3km]$
such that
$\varphi (x)\;:\!=\;\phi ^{-1}_{j}(x)$
if
$x\in I_{j}$
. Let
$ T_{i}$
be the bipartite graph obtained by Lemma 4.12 with vertex classes
$X_{i}\cup Y_{i}$
and
$I_{i}$
, and let
$T=\cup _{i=1}^{k} T_{i}$
. Then
$T$
is a bipartite graph between
$X\cup Y$
and
$[3km]$
with
$\Delta (T)\leq 40$
.
We claim that there exists a family
$\left \{ A_{e} \right \}_{e\in E(T)}$
of pairwise vertex-disjoint subsets in
$V(G)\backslash (X\cup Y\cup Z)$
such that for every
$e=\left \{ w_{1},w_{2} \right \}\in E(T)$
with
$w_{1}\in X\cup Y$
and
$w_{2}\in [3km]$
, the set
$A_{e}$
is a
$(C_{k}, 2k)$
-absorber for the transversal
$k$
-set
$\left \{ w_{1} \right \}\cup \varphi (w_{2})$
. Indeed otherwise, there exists an edge
$e^{\prime }\in E(T)$
without such a subset. Recall that
$m=\frac{qn}{1+\beta ^{\prime }}$
and
$\Delta (T)\leq 40$
, then we have
\begin{equation*} \vert X\vert +\vert Y\vert +\vert Z\vert +\left \vert \bigcup _{e\in E(T)\backslash \left \{ e^{\prime }\right \}} A_{e} \right \vert \leq 4km+3km(k-1)+2k^2\vert E(T) \vert \leq 4k^2m+80k^2\cdot 3km\leq \frac{\tau n}{2}. \end{equation*}
Since every transversal
$k$
-set has at least
$\tau n$
vertex-disjoint
$(C_{k}, 2k)$
-absorbers in
$G$
, we can choose one in
$V(G)\backslash (X \cup Y \cup Z\cup \bigcup _{e\in E(T)\backslash \left \{ e^{\prime }\right \}}A_{e})$
as the subset
$A_{e^{\prime }}$
, a contradiction.
Let
$R=X \cup Y \cup Z\cup \bigcup _{e\in E(T)}A_{e}$
. Then
$\vert R \vert \leq \tau n$
and we claim that
$R$
is a
$\xi$
-absorbing set in
$G$
. Indeed, for an arbitrary balanced subset
$U\subseteq V(G)\backslash R$
with
$\vert U\vert \leq \xi n$
, we shall verify that
$G[R\cup U]$
admits a
$C_k$
-factor. Note that if there exist
$Q_{i}\subseteq X_{i}$
with
$\vert Q_{i} \vert =\beta ^{\prime }m$
for
$i\in [k]$
and a transversal
$C_{k}$
-factor in
$G[\bigcup _{i=1}^{k} Q_{i}\cup U]$
, then
$G[R\cup U]$
contains a transversal
$C_{k}$
-factor. In fact by setting
$X_{i}^{\prime }=X_{i}\backslash Q_{i}$
, Lemma 4.12 implies that there is a perfect matching
$M$
in
$T$
between
$ \bigcup _{i=1}^{k} X_{i}^{\prime }\cup Y$
and
$[3km]$
. For each edge
$e=\left \{ w_{1},w_{2} \right \}\in M$
take a transversal
$C_{k}$
-factor in
$G[\!\left \{ w_{1} \right \}\cup \varphi (w_{2})\cup A_{e}]$
and for each
$e^\prime \in E(T)\backslash M$
take a transversal
$C_{k}$
-factor in
$G[A_{e^\prime }]$
, which forms a transversal
$C_{k}$
-factor of
$G[R\backslash \cup _{i=1}^{k} Q_{i}]$
. Thus together with the above assumption, we can obtain a transversal
$C_{k}$
-factor in
$G[R\cup U]$
.
Hence, it suffices to find the desired
$Q_{i}$
as above. Recall that every
$v\in V(G)$
has a subfamily
$\mathcal{F}_{v}^{\prime }$
of at least
$k\beta ^{\prime }n$
sets in
$\cup _{i=1}^{k} X_{i}$
. By the choice that
$\xi \ll \beta,\frac{1}{k}$
and thus
$\vert U\vert \leq \xi n\leq \beta ^{\prime }n$
, one can greedily find a family
$\mathcal{C}_{1}$
of vertex-disjoint copies of transversal
$C_{k}$
covering
$U$
with vertices in
$\cup _{i=1}^{k}X_{i}$
. Let
$Q_{i1}\;:\!=\;X_{i}\cap V(\mathcal{C}_{1})$
. Then we have
$\vert Q_{i1}\vert =\frac{k-1}{k}\vert U \vert$
and
$\mathcal{C}_1$
is a transversal
$C_{k}$
-factor in
$G[\bigcup _{i=1}^{k} Q_{i1}\cup U]$
. Moreover, one can greedily pick a family
$\mathcal{C}_{2}$
of
$\beta ^{\prime }m-\frac{k-1}{k}\vert U \vert$
vertex-disjoint copies of transversal
$C_{k}$
in
$G[X\backslash V(\mathcal{C}_{1})]$
. This is possible since every vertex
$v$
has at least
$k\beta ^{\prime }n-(k-1)\vert U\vert \geq k(\beta ^{\prime }m-\frac{(k-1)}{k}\vert U \vert )$
sets from
$\mathcal{F}_v$
that are disjoint from
$V(\mathcal{C}_{1})$
. Let
$Q_{i2}\;:\!=\;X_{i}\cap V(\mathcal{C}_{2})$
and
$Q_{i}\;:\!=\;Q_{i1}\cup Q_{i2}$
. Then
$Q_i$
is desired because
$\mathcal{C}_{1}\cup \mathcal{C}_{2}$
is indeed a transversal
$C_k$
-factor of
$G[\!\bigcup _{i=1}^k Q_i\cup U]$
. This completes the entire proof.
4.4.3. Proof of Lemma 4.10
Proof of Lemma
4.10. Given
$k \in \mathbb{N}$
with
$k\geq 4$
and
$\delta \gt \frac{2}{k}$
, we set
$\eta \;:\!=\;\delta -\frac{2}{k}$
and choose
$\frac{1}{n}\ll \alpha \ll \beta \ll \delta,\eta$
. Let
$G=(V_{1},\ldots,V_{k},E)$
be a spanning subgraph of the
$n$
-blow-up of
$C_{k}$
with
$\bar{\delta }(G)\geq \delta n$
and
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n$
.
Claim 4.13.
For each
$i\in [k]$
,
$V_{i}$
is
$(C_{k},\beta n,2)$
-closed.
Proof of Claim
4.13. Without loss of generality, we may assume that
$i=1$
. For any two vertices
$u,v\in V_{1}$
, since
$\bar{\delta }(G)\geq \delta n$
, we can choose four vertex-disjoint sets
$D_{1}$
,
$D_{2}\subseteq V_{2}$
and
$D_{3}, D_{4}\subseteq V_{k}$
each of size at least
$\frac{\delta n}{2}$
such that
$D_{1}\subseteq N_{V_{2}}(u)$
,
$D_{2}\subseteq N_{V_{2}}(v)$
,
$D_{3}\subseteq N_{V_{k}}(u)$
, and
$D_{4}\subseteq N_{V_{k}}(v)$
, respectively. Given any vertex set
$W\subseteq V(G)\backslash \left \{u,v \right \}$
of size at most
$\beta n$
, let
$V^{\prime }_{i}=V_{i}\backslash W$
and
$D^{\prime }_{j}=D_{j}\backslash W$
for
$i\in [k]$
and
$j\in [4]$
. Note that
$\vert V^{\prime }_{i}\vert \geq n-\beta n\geq 10\alpha n$
and
$\vert D^{\prime }_{j}\vert \geq \frac{\delta n}{2}-\beta n\geq \alpha n$
.
Since
$\alpha ^*_{\textrm{b}}(G)\lt \alpha n\leq \vert D_{j}^{\prime }\vert, \vert V^{\prime }_{1}\vert$
, there exist subsets
$S_{j}\subseteq V^{\prime }_{1}$
of size at least
$\vert V^{\prime }_{1}\vert -\alpha n$
such that every vertex in
$S_{j}$
has at least one neighbour inside
$D_{j}^{\prime }$
. By the fact that
$\vert S_{j}\vert \geq \vert V^{\prime }_{1}\vert -\alpha n\gt \frac{3}{4}\vert V^{\prime }_{1}\vert$
, we have that
$S\;:\!=\;\bigcap _{j=1}^{4}S_{j}\neq \emptyset$
. Arbitrarily choose
$x\in S$
, and therefore there exist vertices
$y_{1}$
,
$y_{2}$
,
$y_{3}$
,
$y_{4}$
satisfying
$y_{i}\in N_{D_{i}^{\prime }}(x)$
for
$i\in [k]$
. Note that
$\vert N_{V^{\prime }_{3}}(y_{1})\vert$
,
$\vert N_{V^{\prime }_{k-1}}(y_{3})\vert \geq \delta n-\beta n\geq \alpha n$
and
$\vert V^{\prime }_{i}\vert \geq 10\alpha n$
for
$i\in [k]$
. When
$k\gt 4$
, by applying Proposition 4.7 with
$X_{i}\;:\!=\;N_{V^{\prime }_{3}}(y_{1})$
and
$X_{j}\;:\!=\;N_{V^{\prime }_{k-1}}(y_{3})$
, we can obtain a transversal cycle
$C_{1}$
passing through
$u, y_{1}, y_{3}$
. When
$k=4$
, since
$\vert N_{V^{\prime }_{3}}(y_{1})\vert$
,
$\vert N_{V^{\prime }_{3}}(y_{3})\vert \geq \delta n-\beta n\geq (\frac{1}{2}+\eta -\beta ) n\gt \frac{n}{2}$
, we can easily find a common neighbour of
$y_{1}, y_{3}$
and thus obtain a transversal cycle
$C_{1}$
passing through
$u, y_{1}, y_{3}$
. Similarly, we can obtain a transversal cycle
$C_{2}$
in
$V(G)\backslash (W\cup V(C_{1}))$
that passes through
$v, y_{2}, y_{4}$
. In fact, the set
$\left \{ x \right \}\cup \left (V(C_{1})\backslash \left \{ u \right \}\right )\cup \left (V(C_{2})\backslash \left \{ v \right \}\right )$
is a
$C_{k}$
-connector for
$u$
,
$v$
. Therefore by definition,
$u$
is
$(C_{k},\beta n,2)$
-reachable to
$v$
.
For every transversal
$k$
-subset
$S\subseteq V(G)$
, we greedily find as many pairwise disjoint
$(C_{k},2k)$
-absorbers for
$S$
as possible. For convenience, we write
$S=\left \{ s_{1}, s_{2},\ldots,s_{k} \right \}$
where
$s_{i}\in V_{i}$
for
$i\in [k]$
. Let
$\mathcal{A}=\left \{ A_{1}, A_{2},\ldots,A_{\ell } \right \}$
be a maximal family of such absorbers. Suppose to the contrary that
$\ell \lt \frac{\beta n-k}{4k^2}$
. Since each
$A_{j}$
has size at most
$2k^2$
, we have
$\left \vert \bigcup _{j=1}^{\ell }A_{j}\right \vert \lt \frac{\beta n-k}{2}$
.
Since
$\alpha \ll \beta \ll \delta$
, we can easily find a copy
$T$
of transversal
$C_{k}$
in
$V(G)\backslash (\bigcup _{j=1}^{\ell }A_{j}\cup S)$
and write
$T=\left \{ t_{1}, t_{2},\ldots,t_{k} \right \}$
where
$t_{i}\in V_{i}$
for
$i\in [k]$
. By the closedness of
$V_{i}$
, we can pick a collection
$\left \{ I_{1}, I_{2},\ldots,I_{k} \right \}$
of vertex-disjoint subsets in
$V(G)\backslash (\bigcup _{j=1}^{\ell }A_{j}\cup S\cup T)$
such that each
$I_{i}$
is a
$C_{k}$
-connector for
$s_{i}$
,
$t_{i}$
with
$\vert I_{i} \vert \leq 2k-1$
. In fact, for any
$1\leq k^{\prime }\leq k$
, we have
\begin{equation*} \left \vert \bigcup _{j=1}^{\ell }A_{j}\cup \left (\bigcup _{i=1}^{k^{\prime }}I_{i}\right )\cup S\cup T\right \vert \leq \frac{\beta n-k}{2}+k(2k-1)+2k\lt \beta n. \end{equation*}
Therefore, we can choose such
$I_{i}$
one by one because
$s_{i}$
and
$t_{i}$
are
$(C_{k},\beta n,2)$
-reachable. At this point, it is easy to verify that
$\bigcup _{i=1}^{k} I_{i}\cup T$
is actually a
$(C_{k},2k)$
-absorber for
$S$
, contrary to the maximality of
$\ell$
.
Acknowledgements
We would like to thank the anonymous referees for their thorough reading of the manuscript and many helpful suggestions.
