On the clique-transversal number of chordal graphs Original Research Article

For a graph G, a subgraph C is called a clique of G if C is a complete subgraph of G maximal under inclusion and |bC| ⩾ 2. The clique-transversal number τc(G) is the minimum cardinality of a set of vertices which meets all cliques of G. For k ⩾ 4, let Gk be the class of chordal graphs for which all cliques are either k-cliques (i.e., cliques of order k) or triangles and for which each edge is contained in at least one k-cliques. In response to a question of Tuza, it was shown by Andreae and Flotow (1996) that (i) τc(G)|G| < 27 for all members of a certain subclass G4∗ of G4 and (ii) this bound is best possible, i.e., sup{τc(G)|G|: G ∈ G4∗} = 27. In the present paper, a theorem is presented which extends and generalizes this result. It is shown that τc(G)|G| < min 2(k + 3), 3(2k + 1) for all G ∈ Gk (k ⩾ 4) and a lower bound for σk = supτc(G)|G|: G ∈ Gk is established. In particular, these results show that (σk2k)3 → 1 for k → ∞.