Kai Deng

In this paper, all graphs under consideration are simple, connected and undirected. The line graph $L(G)$ of $G$ is a graph whose vertices are edges of $G$, a pair of vertices in $L(G)$ are adjacent if and only if the correspond two edges are incident in $G$. Obviously, the edge-coloring of $G$ is the same as the vertex-coloring of $L(G)$. The star coloring of $G$ is a proper vertex coloring of $G$ such that any path of length 3 in $G$ is not bicolored, the star chromatic number, denoted by $\chi_{s}(G)$, is the smallest integer $k$ for which $G$ admits a star coloring with $k$ colors. We dene the concept of star-edge coloring on graph, a star-edge coloring of $G$ is a proper edge coloring of $G$ such that any path of length 4 in $G$ is not bicolored, the star chromatic index, denoted by $\chi_{s}'(G)$, is the smallest integer $k$ for which $G$ admits a star-edge coloring with $k$ colors. Guillaume Fertin, Andr′e Raspaud and Bruce Reed proved that if $G$ is a graph with maximum degree $\Delta$ then $\chi_{s}(G)\leq\lceil 20\Delta^{\frac{3}{2}}\rceil$. We will prove that if $\Delta\geq6$ then $\chi_{s}'(G)\leq\lceil 16(\Delta-1)^{\frac{3}{2}}\rceil$, which implies that if G is a line graph with maximum degree $\Delta\geq10$ then $\chi_{s}(G)\leq\lceil16(\Delta-1)^{\frac{3}{2}}\rceil$.