Monochromatic cycle partitions of graphs with large minimum degree

TL;DR

This paper proves a conjecture about partitioning large graphs with high minimum degree into monochromatic cycles, extending previous results from complete graphs to more general graphs.

Contribution

It establishes the validity of a strengthened conjecture on monochromatic cycle partitions in graphs with minimum degree exceeding three-quarters of the number of vertices.

Findings

Proves the conjecture for graphs with minimum degree > (3/4 + o(1))n

Uses Szemerédi's regularity lemma and absorbing method in the proof

Shows such graphs can be covered with monochromatic subgraphs with robust expansion

Abstract

Lehel conjectured that in every $2$-coloring of the edges of $K_n$, there is a vertex disjoint red and blue cycle which span $V(K_n)$. \L uczak, R\"odl, and Szemer\'edi proved Lehel's conjecture for large $n$, Allen gave a different proof for large $n$, and finally Bessy and Thomass\'e gave a proof for all $n$. Balogh, Bar\'at, Gerbner, Gy\'arf\'as, and S\'ark\"ozy proposed a significant strengthening of Lehel's conjecture where $K_n$ is replaced by any graph $G$ with $\delta(G)> 3n/4$; if true, this minimum degree condition is essentially best possible. We prove that their conjecture holds when $\delta(G)>(3/4+o(1))n$. Our proof uses Szemer\'edi's regularity lemma along with the absorbing method of R\"odl, Ruci\'nski, and Szemer\'edi by first showing that the graph can be covered with monochromatic subgraphs having certain robust expansion properties.