On a conjecture about the strong odd chromatic number of planar graphs

TL;DR

This paper investigates the strong odd chromatic number of graphs, specifically focusing on joins of cycles, empty graphs, and unions, and constructs counterexamples to a recent conjecture about planar graphs.

Contribution

It evaluates the strong odd chromatic number for specific graph classes and provides counterexamples to a conjecture on planar graphs' upper bounds.

Findings

Counterexamples to the conjecture on planar graphs.

Exact strong odd chromatic numbers for joins of cycles.

Results on unions of graphs and their chromatic numbers.

Abstract

A proper coloring of a graph $G$ is said to be a strong odd coloring of $G$, if for every vertex $v$ and every color $c$, either $c$ appears on an odd number of vertices in the neighborhood of $v$ or $c$ is absent in the neighborhood of $v$. The strong odd chromatic number of $G$ is defined as the smallest integer $k$ for which $G$ admits a strong odd coloring using $k$ colors. In this paper, we evaluate the strong odd chromatic number of join of cycles and empty graphs and one point union of graphs. Using these results, we construct infinite family of planar graphs that serves as counter examples to a recent conjecture regarding the upper bound of the strong odd chromatic number of planar graphs.