Decomposing random graphs into few cycles and edges

Abstract

Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph $G(n,p)$ with probability approaching 1 as $n\to \infty$. In this paper we show that for most edge probabilities $G(n,p)$ can be decomposed into a union of $\frac{n}{4}+\frac{np}{2}+o(n)$ cycles and edges whp. This result is asymptotically tight.

14 pages

Cited by in corpus (2)

• Optimal path and cycle decompositions of dense quasirandom graphs
• Path and cycle decompositions of dense graphs