The probability that a random graph is even-decomposable

Abstract

A graph $G$ with an even number of edges is called even-decomposable if there is a sequence $V(G)=V_0\supset V_1\supset \dots \supset V_k=\emptyset$ such that for each $i$, $G[V_i]$ has an even number of edges and $V_i\setminus~V_{i+1}$ is an independent set in $G$. The study of this property was initiated recently by Versteegen, motivated by connections to a Ramsey-type problem and questions about graph codes posed by Alon. Resolving a conjecture of Versteegen, we prove that all but an $e^{-\Omega(n^2)}$ proportion of the $n$-vertex graphs with an even number of edges are even-decomposable. Moreover, answering one of his questions, we determine the order of magnitude of the smallest $p=p(n)$ for which the probability that the random graph $G(n,1-p)$ is even-decomposable (conditional on it having an even number of edges) is at least $1/2$. We also study the following closely related property. A graph is called even-degenerate if there is an ordering $v_1,v_2,\dots,v_n$ of its vertices such that each $v_i$ has an even number of neighbours in the set $\{v_{i+1},\dots,v_n\}$. We prove that all but an $e^{-\Omega(n)}$ proportion of the $n$-vertex graphs with an even number of edges are even-degenerate, which is tight up to the implied constant.