Extremal graphs with local covering conditions

TL;DR

This paper investigates the minimal edge count in graphs with fixed vertices where each vertex must belong to a copy of a specific graph H, providing complete solutions for cliques and certain regular graphs, and characterizing extremal graphs for Erdős-Rényi graphs.

Contribution

It completely solves the extremal problem for cliques and regular graphs with high degree, and characterizes extremal graphs for Erdős-Rényi models, advancing understanding of local covering conditions.

Findings

Complete solution for H as a clique.

Solution for regular graphs with degree at least half the vertices.

Characterization of extremal graphs for Erdős-Rényi random graphs.

Abstract

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We completely solve this problem when $H$ is a clique, as well as more generally when $H$ is any regular graph with degree at least about half its number of vertices. We also characterize the extremal graphs when $H$ is an Erd\H{o}s-R\'enyi random graph. The extremal structures turn out to have the similar form as the conjectured extremal structures for a well-studied but elusive problem of similar flavor with local constraints: to maximize the number of copies of a fixed clique in graphs in which all degrees have a fixed upper bound.