3‐Flows and Combs

Abstract

Tutte's 3-Flow Conjecture states that every 2-edge-connected graph with no 3-cuts admits a 3-flow. The 3-Flow Conjecture is equivalent to the following: let G be a 2-edge-connected graph, let S be a set of at most three vertices of G; if every 3-cut of G separates S then G has a 3-flow. We show that minimum counterexamples to the latter statement are 3-connected, cyclically 4-connected, and cyclically 7-edge-connected.