Colorings with neighborhood parity condition

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.