Ilkyoo Choi

1 Introduction Following Grötzsch's Theorem [1], which tells us that planar graphs with no have a proper 3-colouring, Havel [2] conjectured the stronger statement that every planar graph whose s are no closer than some constant also admits a proper 3-colouring. This was proved by Dvořák et al. in [3] with . Steinberg [4] conjectured that every planar graph with neither a 4-cycle nor a 5-cycle is also properly 3-colourable, but a counterexample was found by Cohen-Addad et al. [5].

1 Introduction A -set is a set with elements. All graphs in this paper are finite, simple, and have no loops. Given a graph , let and denote the vertex set and edge set, respectively, of . The odd girth of is the length of a shortest odd cycle in . A proper -coloring of is a function from to such that adjacent vertices receive distinct colors. The chromatic number of , denoted , is the minimum for which has a proper -coloring.