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.

翻译：图$G$的一个\emph{星染色}是一种正常的顶点染色，使得任意四个顶点的路径不是$2$-染色的。对图$G$进行星染色所需的最少颜色数称为星色数，记作$\chi_s(G)$。若图$G$满足$\chi_s(G)=k$，且对每条边$e \in E(G)$有$\chi_s(G -e) < \chi_s(G)$，则称$G$为$k$-临界的。本文中，我们给出了关于星染色的3-临界、$(n-1)$-临界和$(n-2)$-临界图的刻画，其中$n$表示$G$的顶点数。此外，我们还给出了$(n-1)$-临界和$(n-2)$-临界图中最少边数的上界和下界。