Discussiones Mathematicae Graph Theory 33(4) (2013) 677-693
doi: 10.7151/dmgt.1688

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Weak Saturation Numbers for Sparse Graphs

Ralph J. Faudree

Department of Mathematical Sciences
University of Memphis
Memphis, TN 38152

Ronald J. Gould

Department of Math and Computer Science
Emory University
Atlanta, GA 30322

Michael S. Jacobson

Department of Mathematical and Statistical Sciences
University of Colorado Denver
Denver, CO 80217

Abstract

For a fixed graph F, a graph G is F-saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F-saturated graph of order n will be denoted by sat(n, F). A graph G is weakly F-saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs of order n that are weakly F-saturated will be denoted by wSAT(n, F), and those graphs in wSAT(n, F) with wsat(n, F) edges will be denoted by wSAT(n,F). The precise value of wsat(n,T) for many families of sparse graphs, and in particular for many trees, will be determined. More specifically, families of trees for which wsat(n, T) = |T | − 2 will be determined. The maximum and minimum values of wsat(n,T) for the class of all trees will be given. Some properties of wsat(n,T) and wSAT(n,T) for trees will be discussed.

Keywords: saturated graphs, sparse graphs, weak saturation

2010 Mathematics Subject Classification: 05C35.

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Received 27 January 2012
Revised 9 August 2012
Accepted 9 August 2012