Odd coloring of sparse graphs and planar graphs

TL;DR

This paper investigates odd colorings of sparse and planar graphs, resolving a conjecture for most cases, providing counterexamples for some, and establishing new bounds for odd colorings based on graph girth and maximum average degree.

Contribution

The paper fully resolves Cranston's conjecture for odd colorings of graphs with bounded maximum average degree, introduces counterexamples for the case c=4, and establishes new coloring bounds for planar graphs with specific girth.

Findings

Confirmed the conjecture for c ≥ 7

Counterexamples for c=4 containing 5-cycles

Planar graphs with girth ≥ 11 have odd 4-colorings

Abstract

An {\it odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petru\v sevski and \v Skrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph $G$ with $mad(G)\leq \frac{4c-4}{c+1}$ has an odd $c$-coloring for $c\geq 4$, and proved the conjecture for $c\in\{5, 6\}$. In particular, planar graphs with girth at least $7$ and $6$ have an odd $5$-coloring and an odd $6$-coloring, respectively. We completely resolve Cranston's conjecture. For $c\geq 7$, we show that the conjecture is true, in a stronger form that was implicitly suggested by Cranston, but for $c=4$, we construct counterexamples, which all contain $5$-cycles. On the other hand, we show that a graph $G$ with $mad(G)<\frac{22}{9}$ and no induced $5$-cycles has an odd $4$-coloring. This implies that a planar graph with girth at least 11 has an odd $4$-coloring. We also prove that a planar graph with girth at least 5 has an odd $6$-coloring.