ABSTRACT Let denote a (not necessarily properly) edge‐colored balanced bipartite graph on vertices, that is, in which every edge is assigned a color. A cycle in is called properly colored if any two consecutive edges of have distinct colors. A properly colored cycle‐factor of is a spanning subgraph each component of which is a properly colored cycle. We say that is properly colored even vertex‐pancyclic if every vertex of is contained in properly colored cycles of all possible even lengths. Let denote the minimum number of distinct colors appearing on the edges incident with a vertex of . In our first main result we show that implies contains a properly colored cycle of length at most 6. This result can be seen as a specific case regarding the colored version of the Caccetta‐Häggkvist Conjecture for directed graphs and its recently studied bipartite version by Seymour and Spirkl in Combinatorica 4 (2020) 575–599. In our second main result we show that implies that contains a properly colored cycle‐factor. We also show that the color degree bound is essentially sharp. Our final result is the following generalization of a result of Häggkvist and Manoussakis on bipartite tournaments in Combinatorica 9 (1989) 33–38. If is a complete bipartite graph containing no monochromatic paths of length three and has a properly colored Hamilton cycle, then is properly colored even vertex‐pancyclic, unless belongs to two special classes of edge‐colored graphs.
Han et al. (Fri,) studied this question.