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Contractible subgraphs in k-connected graphs

✍ Scribed by Zemin Jin; Xingxing Yu; Xiaoyan Zhang

Publisher: John Wiley and Sons
Year: 2007
Tongue: English
Weight: 185 KB
Volume: 55
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20227

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✦ Synopsis

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 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. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 121–136, 2007

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