Monochromatic Partitions In Local Edge Colorings

Abstract

An edge coloring of a graph is a local r-coloring if the edges incident to any vertex are colored with at most r distinct colors. In this paper, generalizing our earlier work, we study the following problem. Given a family of graphs $\mathcal {F}$ (for example matchings, paths, cycles, powers of cycles and paths, connected subgraphs) and fixed positive integers s, r, at least how many vertices can be covered by the vertices of no more than s monochromatic members of $\mathcal {F}$ in every local r-coloring of $K_n$. Several problems and results are presented. In particular, we prove the following two results. First, if n is sufficiently large then in any local r-coloring of the edges of $K_n$, the vertex set can be partitioned by the vertices of at most r monochromatic trees, which is sharp for local r-colorings (unlike for ordinary r-colorings according to the Ryser conjecture). Second, we show that we can partition the vertex set with at most O(r log r) monochromatic cycles in every local r-coloring of $K_n$. This answers a question of Conlon and Stein and slightly generalizes one of my favorite joint results with Endre (and with Gyárfás and Ruszinkó).