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Paths, Stars and the Number Three

Published online by Cambridge University Press:  12 September 2008

Bruce Reed
Bruce Reed
Affiliation:
Equipe Combinatoire, Institute Blaise Pascal, Université Paris VI, 4 Place Jussieu, Paris 75252, Cedex 05, France Email: reed@lug.ibp.fr
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Abstract

A dominating set for a graph G is a set D of vertices of G such that every vertex of G not in D is adjacent to a vertex of D. We prove that any graph G of minimum degree at least three contains a dominating set D of size at most 3|V(G)|/8. A star S is a graph consisting of a centre x and a set of edges from x to S — x. Clearly, a dominating set D for a graph G corresponds to a set of |D| stars which cover V(G). Thus, we show that the vertices of any graph G of minimum degree 3 can be covered by at most 3|V(G)|/8 vertex disjoint stars. We also show that any connected cubic graph G can be covered by [|V(G)|/9] vertex disjoint paths. Both these results are sharp.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 3 , September 1996 , pp. 277 - 295
Copyright
Copyright © Cambridge University Press 1996

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