Sparse graphs with local covering conditions on edges

TL;DR

This paper characterizes extremal graphs with minimal edges where each edge is contained in a clique of size k or in at least two triangles, generalizing Erdős's original problem and providing solutions for these cases.

Contribution

It provides a complete characterization of extremal graphs for generalized triangle and clique covering conditions, extending Erdős's classical problem.

Findings

Characterization of extremal graphs for clique covering conditions.

Solution to the problem where each edge is in at least two triangles.

Complete resolution of the generalized Erdős problem for these conditions.

Abstract

In 1988, Erd\H{o}s suggested the question of minimizing the number of edges in a connected $n$-vertex graph where every edge is contained in a triangle. Shortly after, Catlin, Grossman, Hobbs, and Lai resolved this in a stronger form. In this paper, we study a natural generalization of the question of Erd\H{o}s in which we replace `triangle' with `clique of order $k$' for ${k\ge 3}$. We completely resolve this generalized question with the characterization of all extremal graphs. Motivated by applications in data science, we also study another generalization of the question of Erd\H{o}s where every edge is required to be in at least $\ell$ triangles for $\ell\ge 2$ instead of only one triangle. We completely resolve this problem for $\ell = 2$.