Odd Coloring of Planar Graphs without Short Cycles

Abstract

An odd coloring of a graph is a proper coloring with the additional constraint that each non-isolated vertex has at least one color that appears an odd number of times in its open neighborhood. Determining if a graph is odd k-colorable is NP-complete for k ≥ 3. It is known that planar graphs are odd 8-colorable and triangle-free planar graphs are odd 7-colorable. In this paper we prove that planar graphs without 4-cycles and 5-cycles, or 4-cycles and 6-cycles, are odd 7-colorable.