Graph cover-saturation

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 $G$ 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.