On coloring of fractional powers of graphs

TL;DR

This paper investigates the chromatic number of fractional powers of graphs, disproving a conjecture in general but confirming it for even powers and providing bounds for odd powers.

Contribution

The paper disproves Iradmusa's conjecture in general and proves it holds for even m, also establishing upper bounds for odd m fractional powers.

Findings

Counterexample for the conjecture when m is odd

Conjecture holds for even m

Upper bound of +2 for odd m when 

Abstract

For $m, n\in \N$, the fractional power $\Gmn$ of a graph $G$ is the $m$th power of the $n$-subdivision of $G$, where the $n$-subdivision is obtained by replacing each edge in $G$ with a path of length $n$. It was conjectured by Iradmusa that if $G$ is a connected graph with $\Delta(G)\ge 3$ and $1<m<n$, then $\chi(\Gmn)=\omega(\Gmn)$. Here we show that the conjecture does not hold in full generality by presenting a graph $H$ for which $\chi(H^{3/5})>\omega(H^{3/5})$. However, we prove that the conjecture is true if $m$ is even. We also study the case when $m$ is odd, obtaining a general upper bound $\chi(\Gmn)\leq \omega(\Gmn)+2$ for graphs with $\Delta(G)\geq 4$.