A Note on Odd Colorings of 1-Planar Graphs

TL;DR

This paper proves that every 1-planar graph can be properly colored with 23 colors such that each non-isolated vertex has a uniquely odd-colored neighbor, extending odd coloring concepts beyond planar graphs.

Contribution

It establishes that all 1-planar graphs admit an odd 23-coloring, advancing the understanding of odd colorings in graphs with crossing edges.

Findings

Every 1-planar graph admits an odd 23-coloring.

Extends odd coloring results from planar to 1-planar graphs.

Provides bounds for odd colorings in graphs with crossings.

Abstract

A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petru\v{s}evski and \v{S}krekovski, who proved that every planar graph admits an odd $9$-coloring; they also conjectured that every planar graph admits an odd $5$-coloring. Shortly after, this conjecture was confirmed for planar graphs of girth at least seven by Cranston; outerplanar graphs by Caro, Petru\v{s}evski, and \v{S}krekovski. Building on the work of Caro, Petru\v{s}evski, and \v{S}krekovski, Petr and Portier then further proved that every planar graph admits an odd $8$-coloring. In this note we prove that every 1-planar graph admits an odd $23$-coloring, where a graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge.