Weak Saturation Numbers for Sparse Graphs

Ralph J. Faudree; Ronald J. Gould; Michael S. Jacobson

Discussiones Mathematicae Graph Theory (2013)

  • Volume: 33, Issue: 4, page 677-693
  • ISSN: 2083-5892

Abstract

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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.

How to cite

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Ralph J. Faudree, Ronald J. Gould, and Michael S. Jacobson. "Weak Saturation Numbers for Sparse Graphs." Discussiones Mathematicae Graph Theory 33.4 (2013): 677-693. <http://eudml.org/doc/267922>.

@article{RalphJ2013,
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.},
author = {Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {saturated graphs; sparse graphs; weak saturation},
language = {eng},
number = {4},
pages = {677-693},
title = {Weak Saturation Numbers for Sparse Graphs},
url = {http://eudml.org/doc/267922},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Ralph J. Faudree
AU - Ronald J. Gould
AU - Michael S. Jacobson
TI - Weak Saturation Numbers for Sparse Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 4
SP - 677
EP - 693
AB - 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.
LA - eng
KW - saturated graphs; sparse graphs; weak saturation
UR - http://eudml.org/doc/267922
ER -

References

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