@MISC{Jin06contractiblesubgraphs, author = {Zemin Jin and Xingxing Yu and Xiaoyan Zhang}, title = {Contractible Subgraphs in k-Connected Graphs }, year = {2006} }

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Abstract

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 k-contractible clique of size at most 3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabayashiâ€™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.