1-planar graphs are odd 13-colorable

TL;DR

This paper proves that every 1-planar graph can be colored with at most 13 colors such that each non-isolated vertex has an odd number of neighbors of each color, improving previous bounds.

Contribution

It establishes that the odd chromatic number of 1-planar graphs is at most 13, refining earlier results that required up to 23 colors.

Findings

Every 1-planar graph is odd 13-colorable.

Improves the upper bound from 23 to 13 colors.

Advances understanding of odd coloring in near-planar graphs.

Abstract

An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $\chi_o(G)$, is the minimum number of colors that admits an odd coloring of $G$. Petru\v{s}evski and \v{S}krekovski in 2021 introduced this notion and proved that if $G$ is planar, then $\chi_o(G)\le9$ and conjectured that $\chi_o(G)\le5$. More recently, Petr and Portier improved $9$ to $8$. 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. Cranston, Lafferty and Song showed that every $1$-planar graph is odd $23$-colorable. In this paper, we improved this result and showed that every $1$-planar graph is odd $13$-colorable.