A Note on Odd Colorings of 1-Planar Graphs
Abstract
A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petrusevski and Skrekovski, who proved that every planar graph admits an odd 9-coloring; they also conjectured that every planar graph admits an odd 5-coloring. Shortly after, this conjecture was confirmed for planar graphs of girth at least seven by Cranston; outerplanar graphs by Caro, Petrusevski, and Skrekovski. Building on the work of Caro, Petrusevski, and Skrekovski, Petr and Portier then further proved that every planar graph admits an odd 8-coloring. In this note we prove that every 1-planar graph admits an odd 23-coloring, where a graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge.