Proper Cycles and Rainbow Cycles in 2-triangle-free edge-colored Complete Graphs

Abstract

An edge-colored graph is called rainbow if every two edges receive distinct colors, and called proper if every two adjacent edges receive distinct colors. An edge-colored graph $G^c$ is properly vertex/edge-pancyclic if every vertex/edge of the graph is contained in a proper cycle of length k for every k with $3\le k\le |V(G)|$. A triangle C of $G^c$ is called a 2-triangle if C receives exactly two colors. We call $G^c$ 2-triangle-free if $G^c$ contains no 2-triangle. Let $d^c(v)$ be the number of colors on the edges incident to v in $G^c$ and let $\delta ^c(G)$ be the minimum $d^c(v)$ over all the vertices $v \in V(G^c)$. We show in this paper that: (i) a 2-triangle-free edge-colored complete graph $K^c_n$ is properly vertex-pancyclic if $\delta ^c(K_n)\ge \lceil \frac{n}{3}\rceil +1$, and is properly edge-pancyclic if $\delta ^c(K_n)\ge \lceil \frac{n}{3}\rceil +2$. (ii) with the exception of a few edge-colored graphs on at most 9 vertices, every vertex of a 2-triangle-free edge-colored complete graph $K^c_n$ with $\delta ^c(K_n)\ge 4$ is contained in a rainbow $C_4$; (iii) every vertex of a 2-triangle-free edge-colored complete graph $K^c_n$ with $\delta ^c(K_n)\ge 5$ is contained in a rainbow $C_5$ unless G is proper or G is a special edge-colored $K_8$.