On strong odd colorings of graphs

TL;DR

This paper investigates the strong odd chromatic number of various graph classes, establishing bounds for planar graphs and exploring colorings of trees, unicyclic graphs, and graph products.

Contribution

It answers whether a universal bound exists for all planar graphs' strong odd chromatic number and extends the study to other graph classes.

Findings

Bound established for planar graphs' strong odd chromatic number.

Characterization of strong odd colorings for trees and unicyclic graphs.

Analysis of strong odd colorings in graph products.

Abstract

A strong odd coloring of a simple graph $G$ is a proper coloring of the vertices of $G$ such that for every vertex $v$ and every color $c$, either $c$ is used an odd number of times in the open neighborhood $N_G(v)$ or no neighbor of $v$ is colored by $c$. The smallest integer $k$ for which $G$ admits a strong odd coloring with $k$ colors is the strong odd chromatic number, $\chi_{soc}(G)$. These coloring notion and graph parameter were recently defined in [H. Kwon and B. Park, Strong odd coloring of sparse graphs, ArXiv:2401.11653v2]. We answer a question raised by the originators concerning the existence of a constant bound for the strong odd chromatic number of all planar graphs. We also consider strong odd colorings of trees, unicyclic graphs and graph products.