1-Planar Graphs Are Odd 13-Colorable

An odd coloring of a graph G is a proper coloring such that any non-isolated vertex in G has a color appearing an odd number of times on its neighbors. The odd chromatic number, denoted by χ o ( G ), is the minimum number of colors that admits an odd coloring of G . Petruševski and Škrekovski in 2021 introduced this notion and proved that if G is planar, then χ o ( G ) ≤ 9 and conjectured that χ o ( G ) ≤ 5. More recently, Petr and Portier improved 9 to 8. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showed that every 1-planar graph is odd 23-colorable. In this paper, we improved this result and showed that every 1-planar graph is odd 13-colorable.