Tutte’s 3-flow Conjecture for almost even graphs

Abstract

A 3-flow for a graph G is an assignment of directions and weights in {1, 2} to the edges of G, such that the netsum of weights on the edges incident to every vertex of G is equal to zero (weights on incoming and outgoing edges are added with opposite signs). Not every graph admits a 3-flow. Aside from graphs with an edge cut of size one, K4 is the simplest graph that does not admit a 3-flow. Tutte’s famous 3-flow Conjecture postulates that every 2-edge-connected graph without edge cuts of size three (3-cuts) admits a 3-flow. A slightly stronger form of this conjecture allows up to three 3-cuts. In this work, our objective is to study classes of graphs with up to four 3-cuts, in fact four vertices of degree three (3-vertices), that admit a 3-flow. We focus on almost even graphs, i.e., graphs with a small number of odd vertices. We obtain a characterization for graphs with up to four odd vertices. We also obtain partial characterizations for graphs with up to four 3-vertices and two odd vertices of higher degree. As expected, 3-vertices play a central role in blocking the existence of 3-flows. Our attempt at understanding this role is motivated by the need to allow a small number of additional 3-vertices in inductive proofs of restricted forms of Tutte’s Conjecture.