Odd-’s in stability critical graphs

A subdivision of K 4 is called an odd- K 4 if each triangle of the K 4 is subdivided to form an odd cycle, and is called a fully odd- K 4 if each of the six edges of the K 4 is subdivided into a path of odd length. A graph G is called stability critical if the deletion of any edge from G increases the stability number. In 1993, Sewell and Trotter conjectured that in a stability critical graph every triple of edges which share a common end is contained in a fully odd- K 4 . The purpose of this note is to show that such a triple is contained in an odd- K 4 .