Partitioning 2-edge-colored graphs by monochromatic paths and cycles

TL;DR

This paper investigates how to partition vertices of 2-edge-colored graphs into monochromatic paths and cycles, proving asymptotic cases of conjectures and establishing new bounds based on graph properties like independence number and minimum degree.

Contribution

It proves asymptotic versions of Sárközy's conjecture and extends results related to Schelp's conjecture, providing new bounds for monochromatic cycle and path partitions in 2-edge-colored graphs.

Findings

Partition into at most 2α(G) monochromatic cycles asymptotically.

Covering almost all vertices with two disjoint monochromatic cycles under high minimum degree.

Covering all but c(H) vertices with two disjoint paths when the complement avoids a bipartite graph H.

Abstract

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2\alpha(G)$ monochromatic cycles, where $\alpha(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1+\eps){3|V(G)|\over 4}$ can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that $\overline{G}$ does not contain a fixed bipartite graph $H$, we show that in every $2$-edge-coloring of $G$, $|V(G)|-c(H)$ vertices can be covered by two vertex disjoint paths of different colors, where $c(H)$ is a constant depending only on $H$. In particular, we prove that $c(C_4)=1$, which is best possible.