Monochromatic cycle partitions in random graphs

TL;DR

This paper extends a classical monochromatic cycle covering result from complete graphs to random graphs, showing that with high probability, such graphs can be covered by a bounded number of monochromatic cycles depending on the number of colors.

Contribution

The authors generalize the monochromatic cycle covering theorem to binomial random graphs, providing bounds that depend on the probability parameter and number of colors.

Findings

High probability coverage of random graphs by monochromatic cycles

Bounded number of cycles depending on r and p

Answers a previously open question in the field

Abstract

Erd\H{o}s, Gy\'arf\'as and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 \log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = \Omega(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 \log r $ vertex-disjoint monochromatic cycles. This answers a question of Kor\'andi, Mousset, Nenadov, \v{S}kori\'{c} and Sudakov.