Monochromatic Cycle Partitions in Local Edge Colorings

Abstract

An edge coloring of a graph is said to be an r-local coloring if the edges incident to any vertex are colored with at most r colors. Generalizing a result of Bessy and Thomasse, we prove that the vertex set of any 2-locally colored complete graph may be partitioned into two disjoint monochromatic cycles of different colors. Moreover, for any natural number r, we show that the vertex set of any r-locally colored complete graph may be partitioned into Or2logr disjoint monochromatic cycles. This generalizes a result of Erdi¾?s, Gyarfas, and Pyber.