Rainbow Connection Number of Sunlet Graph and its Line, Middle and Total Graph

Abstract

A path in an edge−colored graph is said to be a rainbow path if every edge in the path has a different color. An edge colored graph is rainbow connected if there exists a rainbow path between every pair of its vertices. The rainbow connection number of a graph G, denoted by $rc(G)$, is the smallest number of colors required to color the edges of G such that G is rainbow connected. Given two arbitrary vertices u and v in G, a rainbow $u-v$ geodesic in G is a rainbow $u-v$ path of length $d(u, v)$, where $d(u, v)$ is the distance between u and v. G is strongly rainbow connected if there exist a rainbow $u-v$ geodesic for any two vertices u and v in G. The strong rainbow connection number of G, denoted by $src(G)$, is the minimum number of colors required to make G strongly rainbow connected. SyafrizalSyet. al. in \cite{2} proved that, for the sunlet graph $S_n$, $rc(S_n)=src(S_n)=\lfloor\frac{n}{2}\rfloor+n$ for $n\geq2$. In this paper, we improve this result and showthat
$rc(S_n)=src(S_n)=\left\{
\begin{array}{ll}
n, & \hbox{if n is odd;} \\
\frac{3n-2}{2}, & \hbox{if n is even.}
\end{array}
\right.$
We also obtain the rainbow connection number and strong rainbow connection number for the line, middle and total graphs of $S_n$.