On coloring of fractional powers of graphs

For $m,n\in \mathbb{N}$, the fractional power $\text{Gmn}$ 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 $\mathrm{\Delta }(G)\ge 3$ and $1<m<n$, then $\chi (\text{Gmn})=\omega (\text{Gmn})$. Here we show that the conjecture does not hold in full generality by presenting a graph $H$ for which $\chi (H^{3\mathrm{/}5})>\omega (H^{3\mathrm{/}5})$. 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 $\chi (\text{Gmn})\le \omega (\text{Gmn})+2$ for graphs with $\mathrm{\Delta }(G)\ge 4$.