Graph cover-saturation
Danny Rorabaugh

TL;DR
This paper introduces and explores the concept of graph cover-saturation, analyzing its theoretical properties, bounds, and densities for various graph classes, extending classical saturation theory.
Contribution
It defines the new concept of cover-saturation, develops initial structural bounds, and determines asymptotic densities for key graph families.
Findings
Established asymptotic cover-saturation densities for cliques and paths.
Derived upper and lower bounds for cycles and stars with small gaps.
Presented preliminary theory and structural bounds for cover-saturation numbers.
Abstract
Graph is -saturated if contains no copy of graph but any edge added to produces at least one copy of . One common variant of saturation is to remove the former restriction: is -semi-saturated if any edge added to produces at least one new copy of . In this paper we take this idea one step further. Rather than just allowing edges of to be in a copy of , we require it: is -covered if every edge of is in a copy of . 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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Taxonomy
TopicsGraph Theory and Algorithms · Limits and Structures in Graph Theory · Advanced Graph Neural Networks
