Properly colored short cycles in edge-colored graphs

Abstract

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta–Häggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers s and t with t≥s≥2, every edge-colored graph G with no properly colored Ks,t contains a spanning subgraph H which admits an orientation D such that every directed cycle in D is a properly colored cycle in G. Using this result, we show that for r≥4, if the Caccetta–Häggkvist Conjecture holds , then every edge-colored graph of order n with minimum color degree at least n/r+2n+1 contains a properly colored cycle of length at most r. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) Ks,t.