Odd Colorings of Sparse Graphs

TL;DR

This paper investigates the minimum number of colors needed for odd colorings in sparse graphs, providing bounds based on maximum average degree and establishing exact values for specific cases.

Contribution

It introduces bounds on odd chromatic number for graphs with bounded average degree and determines exact values for certain thresholds, advancing understanding of odd colorings in sparse graphs.

Findings

Bounded the odd chromatic number for graphs with maximum average degree less than 4.

Established that hi_o(\u211a_) q 5 and hi_o() q 6.

Provided nearly sharp upper bounds for hi_o() for  in [0,4).

Abstract

A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. The smallest number of colors that admits an odd coloring of a graph $G$ is denoted $\chi_o(G)$. This notion was introduced by Petru\v{s}evski and \v{S}krekovski, who proved that if $G$ is planar then $\chi_o(G)\le 9$; they also conjectured that $\chi_o(G)\le 5$. For a positive real number $\alpha$, we consider the maximum value of $\chi_o(G)$ over all graphs $G$ with maximum average degree less than $\alpha$; we denote this value by $\chi_o(\mathcal{G}_{\alpha})$. We note that $\chi_o(\mathcal{G}_{\alpha})$ is undefined for all $\alpha\ge 4$. In contrast, for each $\alpha\in[0,4)$, we give a (nearly sharp) upper bound on $\chi_o(\mathcal{G}_{\alpha})$. Finally, we prove $\chi_o(\mathcal{G}_{20/7})= 5$ and $\chi_o(\mathcal{G}_3)= 6$. Both of these results are sharp.