Contractible Subgraphs in 3-Connected Graphs

For a graph G we define a graph T (G) whose vertices are the triangles in G and two vertices of T (G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k-connected graph and T (G) contains no edge then G admits a kcontractible clique of size at most 3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabayashi’s result by showing that if G is k-connected and the maximum degree of T (G) is at most 1, then G admits a k-contractible clique of size at most 3 or there exist independent edges e and f of G such that e and f are contained in triangles sharing an edge and G/e/f is k-connected. Xingxing Yu was partially supported by NSF grant DMS-0245530 and NSA grant MDA-904-03-1-0052