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 χo(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 χo(G)≤9 and conjectured that χo(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.