Saturated Graphs of Prescribed Minimum Degree

TL;DR

This paper investigates the minimum number of edges in large, minimum degree constrained graphs that are saturated with respect to complete subgraphs, confirming a conjecture for triangles and providing new bounds.

Contribution

It proves that sat_t(n,p) asymptotically equals tn, confirming Bollobás's conjecture for p=3, and introduces new constructions and hypergraph analogues.

Findings

sat_t(n,p) = tn - O(1) as n→∞

Confirmed Bollobás's conjecture for p=3

Provided new upper bounds and hypergraph extensions

Abstract

A graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but the addition of any new edge to $G$ creates a copy of $H$. In this paper we are interested in the function sat$_{t}(n,p)$, defined to be the minimum number of edges that a $K_{p}$-saturated graph on $n$ vertices can have if it has minimum degree at least $t$. We prove that sat$_{t}(n,p) = tn - O(1)$, where the limit is taken as $n$ tends to infinity. This confirms a conjecture of Bollob\'as when $p = 3$. We also present constructions for graphs that give new upper bounds for sat$_{t}(n,p)$ and discuss an analogous problem for saturated hypergraphs.