Graph Powers and Graph Homomorphisms

TL;DR

This paper explores properties of fractional graph powers, establishing homomorphism equivalences, ordering relations, and a new characterization of circular chromatic number, advancing understanding of graph coloring and homomorphisms.

Contribution

It introduces new homomorphism and ordering results for fractional powers of graphs and provides an alternative definition for circular chromatic number.

Findings

Homomorphism equivalence between fractional powers and inverse powers.

Ordering of fractional powers based on rational parameters.

New characterization of circular chromatic number.

Abstract

In this paper we investigate some basic properties of fractional powers. In this regard, we show that for any rational number $1\leq {2r+1\over 2s+1}< og(G)$, $G^{{2r+1\over 2s+1}}\longrightarrow H$ if and only if $G\longrightarrow H^{-{2s+1\over 2r+1}}.$ Also, for two rational numbers ${2r+1\over 2s+1} < {2p+1\over 2q+1}$ and a non-bipartite graph $G$, we show that $G^{2r+1\over 2s+1} < G^{2p+1\over 2q+1}$. In the sequel, we introduce an equivalent definition for circular chromatic number of graphs in terms of fractional powers. We also present a sufficient condition for equality of chromatic number and circular chromatic number.