For G = G n,1/2 , the Erdős-Renyi random graph, let X n be the random variable representing the number of distinct partitions of V (G) into sets A 1 , . . . , A q so that the degree of each vertex inand if q ≥ 4 is even then X n d − → Po(2 q /q!). More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[A i ] to be congruent to x i modulo q for each i ∈ [q], where the residues x i may be chosen freely. For q = 2, the distribution is not asymptotically Poisson, but it can be determined explicitly.

For G=Gn,1false/2$$ G={G}_{n,1/2} $$, the Erdős–Renyi random graph, let Xn$$ {X}_n $$ be the random variable representing the number of distinct partitions of Vfalse(Gfalse)$$ V(G) $$ into sets A1,…,Aq$$ {A}_1,\dots, {A}_q $$ so that the degree of each vertex in Gfalse[Aifalse]$$ G\left[{A}_i\right] $$ is divisible by q$$ q $$ for all i∈false[qfalse]$$ i\in \left[q\right] $$. We prove that if q≥3$$ q\ge 3 $$ is odd then Xn→dPofalse(1false/q!false)$$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left(1/q!\right) $$, and if q≥4$$ q\ge 4 $$ is even then Xn→dPofalse(2qfalse/q!false)$$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left({2}^q/q!\right) $$. More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in Gfalse[Aifalse]$$ G\left[{A}_i\right] $$ to be congruent to xi$$ {x}_i $$ modulo q$$ q $$ for each i∈false[qfalse]$$ i\in \left[q\right] $$, where the residues xi$$ {x}_i $$ may be chosen freely. For q=2$$ q=2 $$, the distribution is not asymptotically Poisson, but it can be determined explicitly.

A graph is said to be interval colourable if it admits a proper edge-colouring using palette N in which the set of colours incident to each vertex is an interval. The interval colouring thickness of a graph G is the minimum k such that G can be edge-decomposed into k interval colourable graphs. We show that θ(n), the maximum interval colouring thickness of an n-vertex graph, satisfies log(n) 1/3−o(1) θ(n) n 5/6+o(1) , which improves on the trivial lower bound and an upper bound of Axenovich and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan.

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We prove that, for every graph F with at least one edge, there is a constant c F such that there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F -free induced subgraph has chromatic number at most c F . This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Moreover, we show an analogous statement where clique number is replaced by odd girth.