Steinberg's conjecture for higher surfaces

Carl Yerger

Davidson College

Abstract

In this talk, we will describe work related to a conjecture of Steinberg, which states that if a planar graph excludes 4-cycles and 5-cycles, then it is 3-colorable. Our work aims to give baseline results for graphs on higher surfaces. In particular, we show that if G is drawn in surface S, is 4-critical and has no cycles of length four through ten, then the number of vertices in G is at most a constant times the Euler genus of S, where c is an explicit constant that comes out of the proof. This is joint work with Robin Thomas.