On Color Critical Graphs of Star Coloring

A \emph{star coloring} of a graph $G$ is a proper vertex-coloring such that no path on four vertices is $2$-colored. The minimum number of colors required to obtain a star coloring of a graph $G$ is called star chromatic number and it is denoted by $\chi_s(G)$. A graph $G$ is called $k$-critical if $\chi_s(G)=k$ and $\chi_s(G -e) < \chi_s(G)$ for every edge $e \in E(G)$. In this paper, we give a characterization of 3-critical, $(n-1)$-critical and $(n-2)$-critical graphs with respect to star coloring, where $n$ denotes the number of vertices of $G$. We also give upper and lower bounds on the minimum number of edges in $(n-1)$-critical and $(n-2)$-critical graphs.