Monochromatic cycle partitions of graphs with large minimum degree

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 ). Luczak, Rödl, and Szemerédi proved Lehel's conjecture for large n, Allen gave a different proof for large n, and finally Bessy and Thomassé gave a proof for all n.Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed a significant strengthening of Lehel's conjecture where K n is replaced by any graph G with δ(G) > 3n/4; if true, this minimum degree condition is essentially best possible. We prove that their conjecture holds when δ(G) > (3/4 + o(1))n. Our proof uses Szemerédi's regularity lemma along with the absorbing method of Rödl, Ruciński, and Szemerédi by first showing that the graph can be covered with monochromatic subgraphs having certain robust expansion properties. In 1973, Rosta [27] and independently, Faudree and Schelp [11] exactly determined the Ramsey number for all pairs of cycles, which gave an analog of Theorem 1.1 for cycles. Later, this was slightly refined by Faudree, Lesniak, and Schiermeyer [10] to give the following best possible result about long monochromatic cycles.

Monochromatic Cycle Partitions of $2$-Coloured Graphs with Minimum Degree $3n/4$

“…This technique, which we shall describe in more detail in Section 6, has become fairly standard by now and can be used to prove the approximate result of Balogh et al [3]. The second result by DeBiasio and Nelsen [8], requires further ideas, most notably the "absorbing technique" of Rödl, Ruciński and Szemerédi (see [25] and [20]). Nevertheless, the stronger conditions on the minimum degree make their proof a great deal easier than ours.…”

“…Recently, DeBiasio and Nelsen [8] proved the following stronger approximate result of the latter conjecture: for every ε > 0 there exists n 0 such that, for every 2-coloured graph G on n ≥ n 0 vertices and δ(G) ≥ (3/4 + ε)n, the vertex set may be partitioned into two monochromatic cycles of distinct colours.…”

Partitioning 2‐Edge‐Colored Ore‐Type Graphs by Monochromatic Cycles

“…Recently Theorem was improved by DeBiasio and Nelsen who proved that the given minimum degree condition actually implies that there are two vertex disjoint monochromatic cycles of different colors which together cover (i.e., they eliminated the second η in Theorem ).…”

Vertex covers by monochromatic pieces — A survey of results and problems

“…A recent work of DeBiasio and Nelsen [8] introduces a new approach and strengthens Theorem 19 by removing the exceptional set of at most ηn uncovered vertices. Letzer [35] improved that result further, proving the existence of a red-blue cycle partition in every 2-colored n-vertex graph G with δ(G) > 3n 4 , provided that n is large enough.…”