On the clique-transversal number of chordal graphs

Abstract 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 G k 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) τ c (G) |G| 2 7 for all members of a certain subclass G 4∗ of G 4 and (ii) this bound is best possible, i.e., sup { τ c (G) |G| : G ∈ G 4∗} = 2 7 . In the present paper, a theorem is presented which extends and generalizes this result. It is shown that τ c (G) |G| 2 (k + 3) , 3 (2k + 1) for all G ∈ G k (k ⩾ 4) and a lower bound for σ k = sup τ c (G) |G| : G ∈ G k is established. In particular, these results show that (σ k 2k) 3 → 1 for k → ∞.