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 φ of a graph G is said to be odd if for each non-isolated vertex x ∈ V ( G ) there exists a color c such that φ − 1 ( c ) ∩ N ( x ) is odd-sized. We prove that every simple planar graph admits an odd 9-coloring, and conjecture that 5 colors always suffice.