On Coloring Properties of Graph Powers

Abstract

This paper studies some coloring properties of graph powers. We show that $\chi_c(G^{^{\frac{2r+1}{2s+1}}})=\frac{(2s+1)\chi_c(G)}{(s-r)\chi_c(G)+2r+1}$ provided that $\chi_c(G^{^{\frac{2r+1}{2s+1}}})< 4$. As a consequence, one can see that if ${2r+1 \over 2s+1} \leq {\chi_c(G) \over 3(\chi_c(G)-2)}$, then $\chi_c(G^{^{\frac{2r+1}{2s+1}}})=\frac{(2s+1)\chi_c(G)}{(s-r)\chi_c(G)+2r+1}$. In particular, $\chi_c(K_{3n+1}^{^{1\over3}})={9n+3\over 3n+2}$ and $K_{3n+1}^{^{1\over3}}$ has no subgraph with circular chromatic number equal to ${6n+1\over 2n+1}$. This provides a negative answer to a question asked in [Xuding Zhu, Circular chromatic number: a survey, Discrete Math., 229(1-3):371--410, 2001]. Also, we present an upper bound for the fractional chromatic number of subdivision graphs. Precisely, we show that $\chi_f(G^{^{\frac{1}{2s+1}}})\leq \frac{(2s+1)\chi_f(G)}{s\chi_f(G)+1}$. Finally, we investigate the $n$th multichromatic number of subdivision graphs.