Optimal path and cycle decompositions of dense quasirandom graphs

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 $\lfloor \mathrm{\Delta }(G)\mathrm{/}2\rfloor$ cycles and a matching of size $odd(G)\mathrm{/}2$. (ii) $G$ can be decomposed into $\max \{odd(G)\mathrm{/}2,\lceil \mathrm{\Delta }(G)\mathrm{/}2\rceil \}$ paths. (iii) $G$ can be decomposed into $\lceil \mathrm{\Delta }(G)\mathrm{/}2\rceil$ 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.