Properly Colored Cycles in Edge-Colored Balanced Bipartite Graphs

Abstract

Let $G_{n,n}^c$ denote a (not necessarily properly) edge-colored balanced bipartite graph on $2n$ vertices, that is, in which every edge is assigned a color. A cycle $C$ in $G_{n,n}^c$ is called properly colored if any two consecutive edges of $C$ have distinct colors. A properly colored cycle-factor of $G_{n,n}^c$ is a spanning subgraph each component of which is a properly colored cycle. We say that $G_{n,n}^c$ is properly colored even vertex-pancyclic if every vertex of $G_{n,n}^c$ is contained in properly colored cycles of all possible even lengths. Let ${\delta }^c\left(G_{n,n}^c\right)$ denote the minimum number of distinct colors appearing on the edges incident with a vertex of $G_{n,n}^c$. In our first main result we show that ${\delta }^c\left(G_{n,n}^c\right)>n/3+1$ implies $G_{n,n}^c$ 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 ${\delta }^c\left(G_{n,n}^c\right)⩾2n/3+3$ implies that $G_{n,n}^c$ 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 $G_{n,n}^c$ is a complete bipartite graph containing no monochromatic paths of length three and $G_{n,n}^c$ has a properly colored Hamilton cycle, then $G_{n,n}^c$ is properly colored even vertex-pancyclic, unless $G_{n,n}^c$ belongs to two special classes of edge-colored graphs.