Flow-Critical Graphs

In this paper we introduce the concept of k-flow-critical graphs. These are graphs that do not admit a k-flow but such that any smaller graph obtained from it by contraction of edges or of pairs of vertices is k-flowable. Any minimal counter-example for Tutteʹs 3-Flow and 5-Flow Conjectures must be 3-flow-critical and 5-flow-critical, respectively. Thus, any progress towards establishing good characterizations of k-flow-critical graphs can represent progress in the study of these conjectures. We present some interesting properties satisfied by k-flow-critical graphs discovered recently.