On Coloring of graph fractional powers

Abstract

\noindent Let $G$ be a simple graph. For any $k\in N$, the $k-$power of $G$ is a simple graph $G^k$ with vertex set $V(G)$ and edge set $\{xy:d_G(x,y)\leq k\}$ and the $k-$subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. So we can introduce the $m-$power of the $n-$subdivision of $G$, as a fractional power of $G$, that is denoted by $G^{\frac{m}{n}}$. In other words $G^{\frac{m}{n}}:=(G^{\frac{1}{n}})^m$. \noindent In this paper some results about the coloring of $G^{\frac{m}{n}}$ are presented when $G$ is a simple and connected graph and $\frac{m}{n}<1$.