Odd 4-coloring of outerplanar graphs

TL;DR

This paper characterizes exactly which outerplanar graphs are odd 4-colorable, strengthening previous results and providing a complete classification based on the presence of certain blocks.

Contribution

It provides a complete characterization of odd 4-colorable outerplanar graphs, identifying the role of blocks that are not 5-cycles.

Findings

A connected outerplanar graph is odd 4-colorable iff it contains a block not isomorphic to a 5-cycle.

This result strengthens previous bounds on odd colorability of outerplanar graphs.

The characterization is both necessary and sufficient for odd 4-colorability in outerplanar graphs.

Abstract

A proper $k$-coloring of $G$ is called an odd coloring of $G$ if for every vertex $v$, there is a color that appears at an odd number of neighbors of $v$. This concept was introduced recently by Petru\v{s}evski and \v{S}krekovski, and they conjectured that every planar graph is odd 5-colorable. Towards this conjecture, Caro, Petru\v{s}evski, and \v{S}krekovski showed that every outerplanar graph is odd 5-colorable, and this bound is tight since the cycle of length 5 is not odd 4-colorable. Recently, the first author and others showed that every maximal outerplanar graph is odd 4-colorable. In this paper, we show that a connected outerplanar graph $G$ is odd 4-colorable if and only if $G$ contains a block which is not a copy of the cycle of length 5. This strengthens the result by Caro, Petru\v{s}evski, and \v{S}krekovski, and gives a complete characterization of odd 4-colorable outerplanar graphs.