On Coloring of graph fractional powers

TL;DR

This paper investigates the coloring properties of fractional powers of graphs, specifically when the fractional exponent is less than one, expanding understanding of graph coloring in fractional graph transformations.

Contribution

It provides new results on the coloring of fractional powers of graphs, particularly for the case when the fractional exponent is less than one.

Findings

Results on coloring of fractional graph powers for rac{m}{n}<1

Conditions for proper coloring of fractional powers

Extensions of classical coloring results to fractional graph transformations

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 $\frac{m}{n}<1$.