1.1 Main result

In this paper, we study the transversal problem in digraph collections. Let $D$ be a digraph with no loops and at most one edge in each direction between every pair of vertices. Denote the set of vertices of $D$ by $V(D)$ . Define $\delta ^{+}(D)$ and $\delta ^{-}(D)$ as the minimum out-degree and in-degree of $D$ , respectively, and let the minimum semi-degree be $\delta ^0(D) = \min {\delta ^+(D), \delta ^-(D)}$ . For a collection $\mathcal{D} = \{D_1, \ldots , D_m\}$ of digraphs (not necessarily distinct) on a common vertex set $V$ , define $\delta ^0(\mathcal{D}) =\min \{\delta ^0(D_i)\,:\,i\in [m]\}$ .

In 1960, Ghouila-Houri [Reference Ghouila-Houri19] generalised Dirac’s theorem to digraphs.

Chakraborti, Kim, Lee, and Seo [Reference Chakraborti, Kim, Lee and Seo6] suggested that it would be interesting to consider a transversal version of Ghouila-Houri’s theorem. Our main result solves this.

In an undirected graph, one may replace each undirected edge by two arcs with opposite directions. Thus, for sufficiently large $n$ , Theorem1.2 is a direct consequence of Theorem1.4. Furthermore, by setting $D_1 = D_2 = \cdots = D_n$ in Theorem1.4, we immediately obtain Theorem1.3 for all sufficiently large $n$ .

1.2 Notation and organisation

Notation. Throughout the paper, we use standard graph theory notation and terminology. Let $D$ be a digraph with vertex set $V(D)$ and edge set $E(D)$ , and sometimes we write $e(D)\,:\!=\,|E(D)|$ . For $x,y\in V(D)$ , denote by $xy$ the directed edge from $x$ to $y$ and we always write $xy\in D$ . Denote by $P_n$ and $C_n$ the directed path and directed cycle on $n$ vertices, respectively.

We use $H\subseteq D$ to denote that $H$ is a sub-digraph of $D$ . For $S\subseteq V(D)$ , we write $D[S]$ for the sub-digraph of $D$ induced by $S$ and $D-S$ for $D[V(D)\setminus S]$ . For two not necessarily disjoint vertex subsets $A,B\subseteq V(D)$ , let $D[A,B]$ be the sub-digraph of $D$ with vertex set $A\cup B$ and edge set $\{xy\in E(D)\,:\,x\in A, y\in B\}$ and let $e_{D}(A, B)\,:\!=\,|E(D[A,B])|$ . Denote $D^{\pm }[A,B]=D[A,B]\cup D[B,A]$ . When $A=B$ , we often abbreviate $D[A,B]$ and $e_{D}(A, B)$ as $D[A]$ and $e_D(A)$ respectively.

For a digraph $D$ and $v\in V(D)$ , we denote the sets of out-neighbours and in-neighbours of $v$ by $N_D^+(v)$ and $N_D^-(v)$ , respectively, with the corresponding out- and in-degrees given by $d_D^+(v) \,:\!=\, |N_D^+(v)|$ and $d_D^-(v) \,:\!=\, |N_D^-(v)|$ . For $S\subseteq V(D)$ , we write $d_D^+ (v,S)\,:\!=\,|N_D^+(v)\cap S|$ and $d_D^- (v,S)\,:\!=\,|N_D^-(v)\cap S|$ . We often omit subscripts when they are clear from the context.

Let $\mathcal{D}=\{D_i\,:\,i\in [m]\}$ be a collection of digraphs on a common vertex set $V$ and $H$ be a rainbow digraph inside $\mathcal{D}$ . For a vertex $v\in V$ and each $i\in [m]$ , let $N_i^+(v)\,:\!=\,N_{D_i}^+(v)$ , $N_i^-(v)\,:\!=\,N_{D_i}^-(v)$ , $d_i^+(v)\,:\!=\,|N_i^+(v)|$ and $d_i^-(v)\,:\!=\,|N_i^-(v)|$ . For any two vertex subsets $X,\,Y \subseteq V$ , define $\mathcal{D}[X]\,:\!=\,\{D_i[X]\,:\,i\in [m]\}$ , $\mathcal{D}[X,Y]\,:\!=\,\{D_i[X,Y]\,:\,i\in [m]\}$ and $\mathcal{D}^{\pm }[X,Y]\,:\!=\,\{D_i^{\pm }[X,Y]\,:\,i\in [m]\}$ . If $E(D)=\emptyset$ , then we simply write $D= \emptyset$ . Similarly, if there is no edge in $D_i$ for all $i \in [m]$ , then we write $\mathcal{D}= \emptyset$ . Denote by $\textrm {col}(H)$ the set of colours appearing in $H$ . Two rainbow graphs $H_1$ and $H_2$ are said to be disjoint if $V(H_1)\cap V(H_2)=\emptyset$ and $\textrm {col}(H_1)\cap \textrm {col}(H_2)=\emptyset$ .

The notation above extends to undirected graphs in the obvious ways. For $n\in \mathbb{N}$ , we denote the set $\{1,2,\ldots ,n\}$ by $[n]$ . For any two constants $\alpha ,\beta \in (0,1)$ , we write $\alpha \ll \beta$ if there exists a function $f=f(\beta )$ such that the subsequent arguments hold for all $0\lt \alpha \leq f.$

Organisation. We conclude this section with a sketch of the proof of Theorem1.4. In Section 2, we introduce two key techniques that we will need later, i.e., the regularity lemma in digraph collections and a stability result for transversal perfect matchings in bipartite graph collections. For readability, we postpone their proofs to the appendix. The proof of Theorem1.4 will be divided into “stable case” and “extremal case”. Section 3 introduces notions of extremality and stability for digraphs and digraph collections; we prove several results about absorption that will be used in the proof of the stable case. In Section 4, we show that there is always a transversal directed Hamilton cycle in the stable case. Sections 5–7 deal with the extremal case. We finish our paper with some concluding remarks in the final section.

1.3 Discussion of the strategy

Joos and Kim [Reference Joos and Kim21] proved the transversal version of Dirac’s theorem by introducing the auxiliary digraph technique, along with some rotations. However, this technique is not feasible in digraph collections because we cannot extend an existing directed cycle into a longer cycle by using the same rotation. In [Reference Cheng and Staden9], the first author and Staden further developed the absorption method for graph collections, combined with the application of the transversal blow-up lemma, they finally proved a stability version of Joos and Kim’s result [Reference Joos and Kim21]. The absorption technique in [Reference Cheng and Staden9] originates from [Reference Cheng, Wang and Zhao12] and [Reference Cheng, Han, Wang and Wang8], which adapted the absorption technique first introduced in [Reference Rödl, Szemerédi and Ruciński25]. This approach has become a powerful tool for solving various transversal graph embedding problems.

In this paper, we adapt the method developed in [Reference Cheng and Staden9] and generalise their stability from graph collections to digraph collections. By analysing the extremal cases, we proved a tight bound for the transversal Ghouila-Houri’s theorem. We believe that our method could be used to demonstrate stabilities or tight bounds for other transversal graph embedding problems (see the survey [Reference Sun, Wang and Wei26]).

1.4 Sketch of the proof of Theorem 1.4

Our proof utilizes the regularity blow-up method in digraph collections, and is divided into two parts: stable case and extremal case.

Stable case. We first characterise the extremality of a single digraph, which forms the basis of our extremal case distinction. Let $0\lt \frac {1}{n}\ll \epsilon \ll 1$ and $D$ be an $n$ -vertex digraph with minimum semi-degree at least $(\frac {1}{2}-\epsilon )n$ . Suppose $D$ has no directed Hamilton cycle. Then $D$ contains a sub-digraph that is close to one of the digraphs in Figure 1. By Lemma 3.2, such a sub-digraph induces a characteristic partition of $D$ ; we say that $D$ is $(\epsilon , \mathrm{EC}k)$ -extremal for some $k \in [3]$ . Furthermore, this $k$ is unique.

Figure 1.

Extremal digraphs EC1, EC2, and EC3. The grey shaded elliptical indicates that the digraph induced by this vertex set is complete, the grey shaded arrow between two vertex sets indicates that the induced digraph by them is complete in this direction.

Assume that $\mathcal{D} = \{D_1,\ldots , D_n\}$ is a collection of digraphs with the same vertex set $V$ of size $n$ . Suppose that $\delta ^0(\mathcal{D})\geq (\frac {1}{2}-\epsilon )n$ and $\mathcal{D}$ contains no transversal directed Hamilton cycles. We suspect that either almost all digraphs in $\mathcal{D}$ are $(\epsilon ,\textrm {EC}k)$ -extremal for some $k\in [2]$ (allowing $k=1$ for some digraphs and $k=2$ for others), or almost all digraphs in $\mathcal{D}$ are $(\epsilon ,\textrm {EC}3)$ -extremal; moreover those digraphs have similar vertex partitions. Based on this observation, we define two kinds of stability for a digraph collection with minimum degree at least $(\frac {1}{2}-\epsilon )n$ . We say that

• • $\mathcal{D}$ is strongly stable if $\mathcal{D}$ contains many digraphs each of which is not $(\epsilon ,\textrm {EC}k)$ -extremal for any $k\in [3]$ ,

• • $\mathcal{D}$ is weakly stable if for almost all colours $i\in [n]$ , each $D_i$ is $(\epsilon ,\textrm { EC}k)$ -extremal for some $k\in [3]$ , but their vertex partitions are not similar,

• • $\mathcal{D}$ is stable if it is either strongly stable or weakly stable.

In the stable case, we are to show that if $\mathcal{D}$ is stable, then it contains a transversal directed Hamilton cycle.

Step 1. Build an absorbing cycle when $\mathcal{D}$ is stable. (Section 3)

We build a “directed absorbing rainbow cycle” $C$ for $\mathcal{D}$ with the property that $C$ is very small and there is a set $A$ consisting of some colour-vertex-vertex triples $(c,u,v)$ such that whenever $A_0\subseteq A$ is sufficiently small compared to $|C|$ , $C$ can absorb all of its elements. Additionally, we construct an additional auxiliary set $\mathcal{Q}$ consisting of colour-colour-vertex-vertex tuples $(c,c',v,v')$ to absorb those colours and vertices which $C$ cannot. Delete all colours and vertices in $C$ and $\mathcal{Q}$ from $\mathcal{D}$ .

Step 2. Use the regularity-blow-up method for transversals in digraph collections to cover almost all vertices and colours with long rainbow directed paths.

We first prove a regularity lemma for digraph collections (Section 2.1), and then apply it to $\mathcal{D}$ , yielding a reduced digraph collection $\mathcal{R}$ that inherits the minimum degree condition and the stability property of $\mathcal{D}$ . By considering the characteristic bipartite graph of each digraph, we get a bipartite graph collection $\mathcal{B_R}$ for the reduced digraph collection $\mathcal{R}$ . Notice that $\mathcal{R}$ has a disjoint rainbow directed cycle partition if and only if $\mathcal{B_R}$ has a transversal perfect matching. We establish a stability result for transversal perfect matching in bipartite graph collections (Section 2.2), ensuring that the disjoint rainbow directed cycle partition of $\mathcal{R}$ exists. Using the blow-up method, we obtain a set of almost spanning disjoint rainbow directed paths within each directed cycle of $\mathcal{R}$ , which covers almost all vertices outside the absorbing cycle $C$ and the auxiliary set $\mathcal{Q}$ (Section 4).

Step 3. Connect the obtained directed paths and cover the remaining vertices via the absorbing cycle.

In this step, we apply the absorbing property of $C$ as well as the auxiliary set $\mathcal{Q}$ to connect all the directed rainbow paths and to cover the remaining vertices, which ultimately forms a transversal directed Hamilton cycle.

Extremal case (Sections 5–7). Assume that $\delta ^0(\mathcal{D}) \geq \frac {n}{2}$ and $\mathcal{D}$ contains no transversal Hamilton cycles. Let $0\lt \frac {1}{n}\ll \epsilon \ll \delta \ll \eta \ll 1$ . Denote $\mathcal{C}_k\,:\!=\,\{i\in [n]\,:\,D_i\ {\text{is ($\epsilon $,EC$k$)-extremal}}\}$ for $k\in [3]$ and $\mathcal{C}_{\textrm {bad}}\,:\!=\,[n]\setminus (\mathcal{C}_1\cup \mathcal{C}_2\cup \mathcal{C}_3)$ . Notice that $\mathcal{C}_i\cap \mathcal{C}_j=\emptyset$ for $1\leq i\lt j\leq 3$ . By “stable case”, we know that almost all digraphs in $\mathcal{D}$ are extremal and $|\mathcal{C}_1| +|\mathcal{C}_2|\leq 2\sqrt {\delta }n$ if $|\mathcal{C}_3|\geq 2\sqrt {\delta }n$ . In view of Lemma 3.2, we can fix a characteristic partition $({A_i},{B_i},{L_i})$ for each graph $D_i$ with $i\in \mathcal{C}_1\cup \mathcal{C}_2$ , and a characteristic partition $({C_i^1},C_i^2,C_i^3,C_i^4,L_i)$ for each graph $D_i$ with $i\in \mathcal{C}_3$ . By swapping labels, we know that $|{A_1}\triangle {A_i}|,|{B_1}\triangle {B_i}|\lt \delta n$ for every $i\in (\mathcal{C}_1\cup \mathcal{C}_2)\setminus \{1\}$ and $|C_1^k\triangle C_i^k|\lt 2\delta n$ for every $i\in \mathcal{C}_3\setminus \{1\}$ . We proceed with our proof by considering the sizes of $|\mathcal{C}_1|$ , $|\mathcal{C}_2|$ and $|\mathcal{C}_3|$ . We will take the case $|\mathcal{C}_1|\lt \eta n$ , $|\mathcal{C}_2|\geq \eta n$ and $|\mathcal{C}_3|\leq 2\sqrt {\delta }n$ as an example to illustrate the general idea of our proof.

Firstly, expand $A_1\cup B_1$ into an equitable partition $A\cup B$ of $V$ . To apply the transversal blow-up lemma (Lemma 6.2) for finding long rainbow directed paths in $\mathcal{D}$ , we need to ensure most vertices in the partition $A \cup B$ satisfy uniform neighbourhood properties (for example, a vertex $v\in A$ should satisfy $v\in A_i$ for the majority of colours $i\in \mathcal{C}_2$ ). We identify three vertex types in $A$ (and symmetrically in $B$ ) that may disrupt the desired uniformity:

1. Type 1. $v \in A$ belongs to $L_i$ for many $i \in \mathcal{C}_2$ ;

2. Type 2. $v \in A$ belongs to $A_i$ for many $i \in \mathcal{C}_2$ , and $v\in B_j$ for many other $j \in \mathcal{C}_2$ ;

3. Type 3. $v \in A$ belongs to nearly all $B_i$ for $i \in \mathcal{C}_2$ .

We handle these cases as follows: For Type 3 vertices in $A$ , we move them to $B$ (and perform the analogous operation for Type 3 vertices in $B$ ). We define $V_{\textrm {bad}}$ as the union of vertices of Type 1 and Type 2. After this reallocation, every vertex in $A\setminus V_{\textrm {bad}}$ (resp. $B\setminus V_{\textrm {bad}}$ ) lies in $A_i$ (resp. $B_i$ ) for the majority of colours $i\in \mathcal{C}_2$ . However, the partition sizes $|A|$ and $|B|$ may become unbalanced, with their difference bounded by $9\delta n$ . To balance them, we require the following four steps.

Step 1. Balance the number of vertices in $A$ and $B$ .

If there exists a set of disjoint rainbow paths in $\mathcal{D}[B]$ such that after deleting their vertices, the number of vertices in $A$ and $B$ is balanced, then we are done. Otherwise, we will show that $\mathcal{D}$ must contain a transversal directed Hamilton cycle (see Lemma 6.5).

Step 2. Cover vertices in $V_{\textrm {bad}}$ .

Using colours in $\mathcal{C}_2$ , we are to find a sequence of disjoint rainbow directed paths $P_3$ such that their centres are all vertices in $V_{\textrm {bad}}$ and endpoints are unused vertices in $(A\cup B)\setminus V_{\textrm {bad}}$ .

Step 3. Deal with colours in $\mathcal{C}_{\textrm {bad}}\cup \mathcal{C}_3$ .

Choose a maximal rainbow matching $M$ using colours in $\mathcal{C}_{\textrm {bad}}$ and avoiding vertices used in Steps 1-2. It is routine to verify that for each $j \in (\mathcal{C}_{\textrm {bad}}\cup \mathcal{C}_3)\setminus \textrm {col}(M)$ , the graph $D_{j}$ is EC1-extremal.

Step 4. Deal with colours in $(\mathcal{C}_{1}\cup \mathcal{C}_{\textrm {bad}}\cup \mathcal{C}_3)\setminus \textrm {col}(M)$ .

We greedily select two rainbow directed paths using colours from $(\mathcal{C}_{1}\cup \mathcal{C}_{\textrm { bad}}\cup \mathcal{C}_3)\setminus \textrm {col}(M)$ while avoiding the vertices used in the previous three steps.

Based on the minimum degree condition and the characteristic partition of extremal digraphs, one may use colours in $\mathcal{C}_2$ to connect all rainbow paths obtained in the above four steps into a single short rainbow directed path, say $P$ , such that $V_{\textrm {bad}}\subseteq V(P)$ , $\mathcal{C}_1\cup \mathcal{C}_3\cup \mathcal{C}_{\textrm {bad}}\subseteq \textrm {col}(P)$ and $|A\setminus V(P)|$ and $|B\setminus V(P)|$ are nearly equal. By applying the transversal blow-up lemma and some structural analysis, we obtain that $\mathcal{D}$ contains a transversal directed Hamilton cycle.