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Contractible Elements in Graphs and Matroids

Published online by Cambridge University Press:  04 July 2003

HAIDONG WU
Affiliation:
Department of Mathematics, The University of Mississippi, University, MS 38677, USA (e-mail: hwu@olemiss.edu

Abstract

Let G be a simple 3-connected graph with at least five vertices. Tutte [13] showed that G has at least one contractible edge. Thomassen [11] gave a simple proof of this fact and showed that contractible edges have many applications. In this paper, we show that there are at most $\frac{|V(G)|}{5}$ vertices that are not incident to contractible edges in a 3-connected graph G. This bound is best-possible. We also show that if a vertex v is not incident to any contractible edge in G, then v has at least four neighbours having degree three, and each such neighbour is incident to exactly two contractible edges. We give short proofs of several results on contractible edges in 3-connected graphs as well. We also study the contractible elements for k-connected matroids. We partially solve an open problem for regular matroids.

Type
Paper
Copyright
2003 Cambridge University Press

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