On Coloring Properties of Graph Powers

TL;DR

This paper explores coloring properties of graph powers, deriving formulas for circular and fractional chromatic numbers, and providing counterexamples to previous conjectures, with implications for graph coloring theory.

Contribution

It introduces new formulas for circular and fractional chromatic numbers of graph powers and addresses open questions in graph coloring.

Findings

Derived a formula for circular chromatic number of certain graph powers.

Provided a negative answer to a question on subgraph circular chromatic numbers.

Established an upper bound for fractional chromatic number of subdivision graphs.

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.