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 sets of vertices is k-flowable. Throughout this paper we will refer to k-flow-critical graphs simply as k-critical. Any minimum counterexample for Tutte’s 3-Flow and 5-Flow Conjectures must be 3-critical and 5-critical, respectively. Thus, any progress towards establishing good characterizations of k-critical graphs can represent progress in the study of these conjectures. We present some interesting properties satisfied by k-critical graphs discovered recently.