Properly Colored Short Cycles In Edge-Colored Graphs

Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-Haggkvist 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 K-s,(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-Haggkvist Conjecture holds, then every edge-colored graph of order n with minimum color degree at least n/r + 2 root n + 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) K-s,K-t. (C) 2021 Published by Elsevier Ltd.