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

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 Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(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 2edge-colored graph G with minimum degree at least (1+ε)3|V (G)| 4 can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that 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(C4) = 1, which is best possible.1 1 Background, summary of results. In this paper, we consider some conjectures about partitioning vertices of edge-colored graphs into monochromatic cycles or paths. For simplicity, colored graphs means edge-colored graphs in this paper. In this context it is conventional to accept empty graphs and one-vertex graphs as a path or a cycle (of any color) and also any edge as a path or a cycle (in its color). With this convention one can define the cycle (or path) partition number of any colored graph G as the minimum number of vertex disjoint monochromatic cycles (or paths) needed to cover the vertex set of G. For complete graphs, [6] posed the following conjecture. Conjecture 1.1. The cycle partition number of any t-colored complete graph Kn is t. The t = 2 case of this conjecture was stated earlier by Lehel in a stronger form, requiring that the colors of the two cycles must be different. After some initial results [2, 8], Ã Luczak, Rödl and Szemerédi [22] proved Lehel’s conjecture for large ∗Research supported in part by NSF CAREER Grant DMS-0745185, UIUC Campus Research Board Grant 11067, and OTKA Grant K 76099. †Research is supported by OTKA Grants PD 75837 and K 76099, and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Present address: School of Mathematical Sciences, Monash University, 3800 Victoria. ‡Research supported in part by NSF Grant DMS-0968699. Part of the research reported in this paper was done at the 3rd Emléktábla Workshop (2011) in Balatonalmádi, Hungary.