The minimum color degree and a large rainbow cycle in an edge-colored graph

Abstract: Let $G$ be an edge-colored graph with $n$ vertices. A subgraph $H$ of $G$ is called a rainbow subgraph of $G$ if the colors of each pair of the edges in $E(H)$ are distinct. We define the minimum color degree of $G$ to be the smallest number of the colors of the edges that are incident to a vertex $v$, for all $v\in V(G)$. Suppose that $G$ contains no rainbow-cycle subgraph of length four. We show that if the minimum color degree of $G$ is at least $\frac{n+3k-2}{2}$, then $G$ contains a rainbow-cycle subgraph of length at least $k$, where $k\geq 5$. Moreover, if the condition of $G$ is restricted to a triangle-free graph that contains a rainbow path of length at least $\frac{3k}{2}$, then the lower bound of the minimum color degree of $G$ that guarantees an existence of a rainbow-cycle subgraph of length to at least $k$ can be reduced to $\frac{2n+3k-1}{4}$.