Odd 4-Coloring of Outerplanar Graphs

A proper k-coloring of G is called an odd coloring of G if for every vertex v, there is a color that appears at an odd number of neighbors of v. This concept was introduced recently by Petruševski and Škrekovski, and they conjectured that every planar graph is odd 5-colorable. Towards this conjecture, Caro, Petruševski, and Škrekovski showed that every outerplanar graph is odd 5-colorable, and this bound is tight since the cycle of length 5 is not odd 4-colorable. Recently, the first author and others showed that every maximal outerplanar graph is odd 4-colorable. In this paper, we show that a connected outerplanar graph G is odd 4-colorable if and only if G contains a block which is not a copy of the cycle of length 5. This strengthens the result by Caro, Petruševski, and Škrekovski, and gives a complete characterization of odd 4-colorable outerplanar graphs.