The Distance Coloring of Graphs

Let $G$ be a connected graph with maximum degree $Δ\ge 3$. We investigate the upper bound for the chromatic number $χ_γ(G)$ of the power graph $G^γ$. It was proved that $χ_γ(G) \leΔ\frac{(Δ-1)^γ-1}{Δ-2}+1=:M+1$ 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 $g(G) \le 2γ-1$, (3) $g(G) \ge 2γ+2$, and the connectivity $κ(G) \ge 3$ if $γ\ge 3$, $κ(G) \ge 4$ but $g(G) >6$ if $γ=2$, (4) $Δ$ is sufficiently large than a given number only depending on $γ$, then $χ_γ(G) \le M-1$. By means of the spectral radius $λ_1(G)$ of the adjacency matrix of $G$, it was shown that $χ_2(G) \le λ_1(G)^2+1$, with equality holds if and only if $G$ is a star or a Moore graph with diameter 2 and girth 5, and $χ_γ(G) < λ_1(G)^γ+1$ if $γ\ge 3$.

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