Strong rainbow coloring of unicyclic graphs

Abstract

A path in an edge-colored graph is called a textit{rainbow path}, if no two edges of the path are colored the same. An edge-colored graph $G$, is textit{rainbow-connected} if any two vertices are connected by a rainbow path. A rainbow-connected graph is called strongly rainbow connected if for every two distinct vertices $u$ and $v$ of $V(G)$, there exists a rainbow path $P$ from $u$ to $v$ that in the length of $P$ is equal to $d(u,v)$. The notations {rm rc}$(G)$ and {rm src}$(G)$ are the smallest number of colors that are needed in order to make $G$ rainbow connected and strongly rainbow connected, respectively. In this paper, we find the exact value of {rm rc}$(G)$, where $G$ is a unicyclic graph. Moreover, we determine the upper and lower bounds for {rm src}$(G)$, where $G$ is a unicyclic graph, and we show that these bounds are sharp.