Optimal-size clique transversals in chordal graphs

TL;DR

This paper proves that in certain chordal graphs where every edge is in a large clique, there exists a small set of vertices intersecting all maximal cliques, improving previous bounds and establishing optimality.

Contribution

It establishes the best possible upper bound for the size of clique transversals in chordal graphs with large cliques, answering a longstanding open question.

Findings

Clique transversals of size at most 2(n-1)/7 are guaranteed.

The bound is proven to be optimal.

Addresses a question posed by Tuza and Erdős et al.

Abstract

The following question was raised by Tuza in 1990 and Erdos et al. in 1992: if every edge of an n-vertex chordal graph G is contained in a clique of size at least four, does G have a clique transversal, i.e., a set of vertices meeting all non-trivial maximal cliques, of size at most n/4? We prove that every such graph G has a clique transversal of size at most 2(n-1)/7 if n>=5, which is the best possible bound.