Colorings with neighborhood parity condition

TL;DR

This paper introduces a new vertex coloring called odd coloring, motivated by odd edge-colorings, and proves that every simple planar graph can be colored with 9 colors under this scheme, conjecturing that only 5 are needed.

Contribution

The paper defines a novel vertex coloring based on neighborhood parity and proves an upper bound for planar graphs, proposing a conjecture to improve this bound.

Findings

Every simple planar graph admits an odd 9-coloring.

Conjecture that 5 colors suffice for all planar graphs.

Introduction of a new neighborhood parity-based coloring concept.

Abstract

In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. We prove that every simple planar graph admits an odd $9$-coloring, and conjecture that $5$ colors always suffice.