Saturation problems with regularity constraints

TL;DR

This paper investigates the existence and properties of regular graphs that are saturated with respect to a fixed subgraph $F$, focusing on complete graphs and exploring relaxed variants of the saturation problem.

Contribution

It establishes the existence of large regular $F$-saturated graphs for complete graphs and introduces relaxed problem variants, expanding understanding of saturation constraints.

Findings

Existence of $K_3$-saturated regular graphs for large $n$

Analysis of relaxed saturation conditions

Characterization of minimal edge counts in saturated graphs

Abstract

For a graph $F$, we say that another graph $G$ is $F$-saturated, if $G$ is $F$-free and adding any edge to $G$ would create a copy of $F$. We study for a given graph $F$ and integer $n$ whether there exists a regular $n$-vertex $F$-saturated graph, and if it does, what is the smallest number of edges of such a graph. We mainly focus on the case when $F$ is a complete graph and prove for example that there exists a $K_3$-saturated regular graph on $n$ vertices for every large enough $n$. We also study two relaxed versions of the problem: when we only require that no regular $F$-free supergraph of $G$ should exist or when we drop the $F$-free condition and only require that any newly added edge should create a new copy of $F$.