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arXiv · 2112.13710

Colorings with neighborhood parity condition

Mirko Petruševski·Riste Škrekovski
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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 $φ$ 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 $φ^{-1}(c)\cap N(x)$ is odd-sized. We prove that every simple planar graph admits an odd $9$-coloring, and conjecture that $5$ colors always suffice.

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Mirko Petruševski, Riste Škrekovski. 2022-06-29. Colorings with neighborhood parity condition. https://arxiv.org/abs/2112.13710

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