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Graph Cover-Saturation

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Abstract

Graph G is F-saturated if G contains no copy of graph F but any edge added to G produces at least one copy of F. One common variant of saturation is to remove the former restriction: G is F-semi-saturated if any edge added to G produces at least one new copy of F. In this paper we take this idea one step further. Rather than just allowing edges of G to be in a copy of F, we require it: G is F-covered if every edge of G is in a copy of F. It turns out that there is smooth interaction between coverage and semi-saturation, which opens for investigation a natural analogue to saturation numbers. Therefore we present preliminary cover-saturation theory and structural bounds for the cover-saturation numbers of graphs. We also establish asymptotic cover-saturation densities for cliques and paths, and upper and lower bounds (with small gaps) for cycles and stars.

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Acknowledgements

We would like to thank Mike Ferrara and Claude Tardif for helpful suggestions.

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  1. University of Tennessee, Knoxville, USA

    Danny Rorabaugh

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  1. Danny Rorabaugh
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Correspondence to Danny Rorabaugh.

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Rorabaugh, D. Graph Cover-Saturation. Graphs and Combinatorics 35, 1225–1237 (2019). https://doi.org/10.1007/s00373-019-02071-w

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