Bobby
Miraftab
From Erdős’s Triangle Question to Clique Cover Problems
Abstract.
This talk is about connected graphs in which every edge lies in one or more copies of a clique, with a focus on extremal and algorithmic questions. Motivated by Erdős’s question on connected graphs where every edge belongs to a triangle, we survey our recent results on the minimum number of edges in such graphs, characterize extremal constructions. We then turn to completion problems, where the goal is to add as few non-edges as possible so that every edge is covered by a clique, highlighting hardness results, exact algorithms for trees and chordal graphs, and approximation results for broader classes.