Edge-Coloring Planar Graphs and the Cycling Conjecture

Abstract

An r-graph is defined to be a graph where each vertex has degree r and any odd subset has at least r edges leaving it. This thesis focuses on the conjecture that any planar r-graph can be edge-colored with r colors. Past work has shown that the conjecture holds for r ≤ 8, but it becomes more difficult with each increase of r. We consider what occurs when r is very large. The main ideas of the thesis work with a minimal counterexample graph that is one of the smallest graphs to contradict the conjecture for a given r. To make a minimal counterexample easier to work with, we generalize to grafts, working with T-joins and T-cuts. We go through various directions to approach the problem and show properties of a minimal counterexample as well as questions that stand in the way of proving it does not exist.