Rainbow cycles in triangle-free graphs

Abstract: Let $G = (V,E)$ be an edge-colored graph, and let $$δ^c(G) = \min_{v \in V} { d^{c}(v) }$$ where $d<sup>c(v)$ is the number of colors on edges incident to a vertex $v$. We show that for a sufficiently large $n$ if $G$ is an edge-colored triangle-free graph of order $n$ that satisfies $δ^c(G)\geq (n+7)/5$, then $G$ contains a rainbow cycle of length four, which improves a bound of Ding et al. and is best possible. In addition, we show that given $k$, there is $n_0$ such that for $n\geq n_0$, if $G$ is an edge-colored triangle-free graph with $δ^c(G)&gt; n/5+3$, then $G$ contains a rainbow cycle of length $4k$.