Optimal path and cycle decompositions of dense quasirandom graphs

TL;DR

This paper establishes optimal decompositions of dense random and quasirandom graphs into paths, cycles, and forests, confirming longstanding conjectures and extending known results using Hamilton decomposition techniques.

Contribution

It proves optimal decomposition results for dense random graphs into paths, cycles, and forests, and determines the edge chromatic number of dense quasirandom graphs, advancing graph decomposition theory.

Findings

Decomposition into cycles and matchings based on maximum degree and odd vertices.

Decomposition into paths using maximum of odd degree vertices and maximum degree.

Decomposition into linear forests matching maximum degree considerations.

Abstract

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the number of odd degree vertices in $G$. Then a.a.s. the following hold: (i) $G$ can be decomposed into $\lfloor\Delta(G)/2\rfloor$ cycles and a matching of size $odd(G)/2$. (ii) $G$ can be decomposed into $\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\}$ paths. (iii) $G$ can be decomposed into $\lceil\Delta(G)/2\rceil$ linear forests. Each of these bounds is best possible. We actually derive (i)--(iii) from `quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by K\"uhn and Osthus.