Properly colored even cycles in edge-colored complete balanced bipartite graphs

TL;DR

This paper investigates the existence of properly colored even cycles in edge-colored complete bipartite graphs, establishing conditions under which such graphs contain cycles of all even lengths through every vertex.

Contribution

It introduces new minimum color degree conditions ensuring the presence of properly colored even cycles and proves proper vertex-even-pancyclicity for large graphs.

Findings

Graphs with high minimum color degree contain large properly colored 2-factors.

Under certain conditions, large graphs are properly vertex-even-pancyclic.

The results use probabilistic and absorbing techniques to establish cycle existence.

Abstract

Consider a complete balanced bipartite graph $K_{n,n}$ and let $K^c_{n,n}$ be an edge-colored version of $K_{n,n}$ that is obtained from $K_{n,n}$ by having each edge assigned a certain color. A subgraph $H$ of $K^c_{n,n}$ is called properly colored (PC) if every two adjacent edges of $H$ have distinct colors. $K_{n,n}^c$ is called properly vertex-even-pancyclic if for every vertex $u\in V(K_{n,n}^c)$ and for every even integer $k$ with $4 \leq k \leq 2n$, there exists a PC $k$-cycle containing $u$. The minimum color degree $\delta^c(K^c_{n,n})$ of $K^c_{n,n}$ is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on the edges incident to $v$. In this paper we study the existence of PC even cycles in $K_{n,n}^c$. We first show that, for every integer $t\geq 3$, every $K^c_{n,n}$ with $\delta^c(K^c_{n,n})\geq \frac{2n}{3}+t$ contains a PC 2-factor $H$ such that every cycle of $H$ has a length of at least $t$. By using the probabilistic method and absorbing technique, we use the above result to further show that, for every $\varepsilon>0$, there exists an integer $n_0(\varepsilon)$ such that every $K^c_{n,n}$ with $n\geq n_0(\varepsilon)$ is properly vertex-even-pancyclic, provided that $\delta^c(K^c_{n,n})\geq (\frac{2}{3}+\varepsilon)n$.