The Distance Coloring of Graphs

Let $G$ be a connected graph with maximum degree $\Delta \ge 3$. We investigate the upper bound for the chromatic number $\chi_\gamma(G)$ of the power graph $G^\gamma$. It was proved that $\chi_\gamma(G) \le\Delta\frac{(\Delta-1)^{\gamma}-1}{\Delta-2}+1=:M+1$ with equality if and only $G$ is a Moore graph. If $G$ is not a Moore graph, and $G$ holds one of the following conditions: (1) $G$ is non-regular, (2) the girth $g(G) \le 2\gamma-1$, (3) $g(G) \ge 2\gamma+2$, and the connectivity $\kappa(G) \ge 3$ if $\gamma \ge 3$, $\kappa(G) \ge 4$ but $g(G) >6$ if $\gamma =2$, (4) $\Delta$ is sufficiently large than a given number only depending on $\gamma$, then $\chi_\gamma(G) \le M-1$. By means of the spectral radius $\lambda_1(G)$ of the adjacency matrix of $G$, it was shown that $\chi_2(G) \le \lambda_1(G)^2+1$, with equality holds if and only if $G$ is a star or a Moore graph with diameter 2 and girth 5, and $\chi_\gamma(G) < \lambda_1(G)^\gamma+1$ if $\gamma \ge 3$.