On odd colorings of planar graphs

Abstract

An odd k-coloring of a graph G is a proper k-coloring such that every non-isolated vertex in G has a color appearing odd times on its neighborhood. In 2022, Petruševski and Škrekovski [Colorings with neighborhood parity condition, Discrete Appl. Math., 321: 385-391] introduced this concept and conjectured that every planar graph admits an odd 5-coloring. We prove that (1) every planar graph without adjacent 3-cycles is odd 7-colorable; (2) If G is a planar graph without 3-cycles and intersecting 4-cycles, then G is odd 5-colorable; (3) The odd chromatic number of a Halin graph is either 3 or 4.