On Colorings of Graph Powers

TL;DR

This paper investigates graph power colorings, introduces helical graphs, and explores homomorphisms to odd cycles, providing new insights into graph coloring and hom-universality related to high girth graphs.

Contribution

It introduces helical graphs and establishes their hom-universality, and characterizes homomorphisms to odd cycles via powers of subdivided graphs, advancing understanding of graph coloring complexities.

Findings

Helical graphs are hom-universal for high odd-girth graphs.

Homomorphism to odd cycles characterized by powers of subdivided graphs.

Results relate to Nešetřil's Pentagon problem and high girth cubic graphs.

Abstract

In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose $(2t+1)$st power is bounded by a Kneser graph. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a $(2k+1)$-cycle exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$ is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. We also consider Ne\v{s}et\v{r}il's Pentagon problem. This problem is about the existence of high girth cubic graphs which are not homomorphic to the cycle of size five. Several problems which are closely related to Ne\v{s}et\v{r}il's problem are introduced and their relations are presented.