There are several known results asserting that undirected graphs can be partitioned in a way that satisfies various constraints imposed on the degrees. The corresponding results for directed graphs, where degrees are replaced by outdegrees, often fail, and when they do hold, they are usually much harder, and lead to fascinating open problems. In this note we list three problems of this type, and mention the undirected analogues. All graphs and digraphs considered here are simple, that is, they have no loops and no multiple edges.

For a fixed graph $H$, we define the rainbow Turán number $\ex^*(n,H)$ to be the maximum number of edges in a graph on $n$ vertices that has a proper edge-colouring with no rainbow $H$. Recall that the (ordinary) Turán number $\ex(n,H)$ is the maximum number of edges in a graph on $n$ vertices that does not contain a copy of $H$. For any non-bipartite $H$ we show that $\ex^*(n,H)=(1+o(1))\ex(n,H)$, and if $H$ is colour-critical we show that $\ex^{*}(n,H)=\ex(n,H)$. When $H$ is the complete bipartite graph $K_{s,t}$ with $s \leq t$ we show $\ex^*(n,K_{s,t}) = O(n^{2-1/s})$, which matches the known bounds for $\ex(n,K_{s,t})$ up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound $\ex^*(n,C_6) = O(n^{4/3})$, which is of the correct order of magnitude.

Let $P(G,t)$ and $F(G,t)$ denote the chromatic and flow polynomials of a graph $G$. G. D. Birkhoff and D C. Lewis showed that, if $G$ is a plane near-triangulation, then the only zeros of $P(G,t)$ in $(-\infty,2]$ are 0, 1 and 2. We will extend their theorem by showing that a stronger result to the dual statement holds for both planar and non-planar graphs: if $G$ is a bridge graph with at most one vertex of degree other than three, then the only zeros of $F(G,t)$ in $(-\infty,\alpha]$ are 1 and 2, where $\alpha\approx 2.225\cdots$ is the real zero in $(2,3)$ of the polynomial $t^4-8t^3+22t^2-28t+17$. In addition we construct a sequence of ‘near-cubic’ graphs whose flow polynomials have zeros converging to $\alpha$ from above.

We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is $C_G$. We show that if it is finite for some Cayley graph of the group then it is finite for any (finitely generated) Cayley graph. Having an exponential bound for the number of minimal cutsets of size $n$ separating $o$ from infinity also turns out to be independent of the Cayley graph chosen. We show a 1-ended example (the lamplighter group), where $C_G$ is infinite. Finally, we give a new proof for a question of de la Harpe, proving that the number of $n$-element cutsets separating $o$ from infinity is finite unless $G$ is a finite extension of $\mathbb{Z}$.

By the complexity of a graph we mean the minimum number of union and intersection operations needed to obtain the whole set of its edges starting from stars. This measure of graphs is related to the circuit complexity of boolean functions.

We prove some lower bounds on the complexity of explicitly given graphs. This yields some new lower bounds for boolean functions, as well as new proofs of some known lower bounds in the graph-theoretic framework. We also formulate several combinatorial problems whose solution would have intriguing consequences in computational complexity.

For an integer $b \geq 1$, the $b$-choice number of a graph $G$ is the minimum integer $k$ such that, for every assignment of a set $S(v)$ of at least $k$ colours to each vertex $v$ of $G$, there is a $b$-set colouring of $G$ that assigns to each vertex $v$ a $b$-set $B(v) \subseteq S(v) \; (|B(v)|=b)$ so that adjacent vertices receive disjoint $b$-sets. This is a generalization of the notions of choice number and chromatic number of a graph. Using probabilistic arguments, we show that, for some positive constant $c > 0$ (independent of $b$), the $b$-choice number of any graph $G$ on $n$ vertices is at most $c (b\chi) (\ln (n/\chi)+1)$ where $\chi = \chi(G)$ denotes the chromatic number of $G$. For any fixed $b$, this bound is tight up to a constant factor for each $n,\chi$. This generalizes and extends a result of Noga Alon [1]wherein a similar bound was obtained for 1-choice numbers of complete $\chi$-partite graphs with each part having size $n/\chi$. We also show that the proof arguments are constructive, leading to polynomial time algorithms for the list colouring problem on certain classes of graphs, provided each vertex is given a list of sufficiently large size.

We derive the distribution of the number of links and the average weight for the shortest path tree (SPT) rooted at an arbitrary node to $m$ uniformly chosen nodes in the complete graph of size $N$ with i.i.d. exponential link weights. We rely on the fact that the full shortest path tree to all destinations (ie, $m=N-1$) is a uniform recursive tree to derive a recursion for the generating function of the number of links of the SPT, and solve this recursion exactly.

The explicit form of the generating function allows us to compute the expectation and variance of the size of the subtree for all $m$. We also obtain exact expressions for the average weight of the subtree.

Let $H$ be a fixed graph on $h$ vertices. We say that a graph $G$ is induced$H$-free if it does not contain any induced copy of $H$. Let $G$ be a graph on $n$ vertices and suppose that at least $\epsilon n^2$ edges have to be added to or removed from it in order to make it induced $H$-free. It was shown in [5] that in this case $G$ contains at least $f(\epsilon,h)n^h$ induced copies of $H$, where $1/f(\epsilon,h)$ is an extremely fast growing function in $1/\epsilon$, that is independent of $n$. As a consequence, it follows that for every $H$, testing induced $H$-freeness with one-sided error has query complexity independent of $n$. A natural question, raised by the first author in [1], is to decide for which graphs $H$ the function $1/f(\epsilon,H)$ can be bounded from above by a polynomial in $1/\epsilon$. An equivalent question is: For which graphs $H$ can one design a one-sided error property tester for testing induced $H$-freeness, whose query complexity is polynomial in $1/\epsilon$? We settle this question almost completely by showing that, quite surprisingly, for any graph other than the paths of lengths 1,2 and 3, the cycle of length 4, and their complements, no such property tester exists. We further show that a similar result also applies to the case of directed graphs, thus answering a question raised by the authors in [9]. We finally show that the same results hold even in the case of two-sided error property testers. The proofs combine combinatorial, graph-theoretic and probabilistic arguments with results from additive number theory.

The theorems of Hindman and van der Waerden belong to the classical theorems of partition Ramsey Theory. The Central Sets Theorem is a strong simultaneous extension of both theorems that applies to general commutative semigroups. We give a common extension of the Central Sets Theorem and Ramsey's theorem.

Every node of an undirected connected graph is coloured white or black. Adjacent nodes can be compared and the outcome of each comparison is either 0 (same colour) or 1 (different colours). The aim is to discover a node of the majority colour, or to conclude that there is the same number of black and white nodes. We consider randomized algorithms for this task and establish upper and lower bounds on their expected running time. Our main contribution are lower bounds showing that some simple and natural algorithms for this problem cannot be improved in general.

We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomized approximation scheme for the case in which the system is consistent in the sense that the local external fields all favour the same spin. We characterize the complexity of the general problem by showing that it is equivalent in complexity to the problem of approximately counting independent sets in bipartite graphs, thus it is complete in a logically defined subclass of #P previously studied by Dyer, Goldberg, Greenhill and Jerrum. By contrast, we show that the corresponding computational task for the $q$-state Potts model with local external magnetic fields and $q>2$ is complete for all of #P with respect to approximation-preserving reductions.

Let $P_1, {\ldots}\,,P_k$ be $k$ vertex-disjoint paths in a graph $G$ where the ends of $P_i$ are $x_i$, and $y_i$. Let $H$ be the subgraph induced by the vertex sets of the paths. We find edge bounds $E_1(n)$, $E_2(n)$ such that:

1. if $e(H) \geq E_1(|V(H)|)$, then there exist disjoint paths $P_1', {\ldots}\,,P_k'$ where the ends of $P_i'$ are $x_i$ and $ y_i$ such that $|\bigcup_i V(P_i)| > |\bigcup_i V(P_i')|$;

2. if $e(H) \geq E_2(|V(H)|)$, then there exist disjoint paths $P_1', {\ldots}\,, P_k'$ where the ends of $P_i'$ are $x_i'$ and $y_i'$ such that $|\bigcup_i V(P_i)| > |\bigcup_i V(P_i')|$ and $\{ x_1, {\ldots}\,, x_k \} = \{ x_1' , {\ldots}\, , x_k' \}$ and $\{ y_1, {\ldots}\,, y_k \} = \{ y_1', {\ldots}\,, y_k'\}$.

The bounds are the best possible, in that there exist arbitrarily large graphs $H'$ with $e(H') = E_i (H') - 1$ without the properties stipulated in 1 and 2.