On covering all cliques of a chordal graph

For a graph G = (V,E), a vertex set X subset of or equal to V is called a clique if \X\ greater than or equal to 2 and the graph G[X] induced by X is a complete subgraph maximal under inclusion. We say that a vertex set T is a clique-transversal set if T boolean AND X not equal 0 for all cliques X of G, and define the clique-transversal number tau(c)(G) as the minimum cardinality of a clique-transversal set. Let G be the class of chordal graphs with the property that each edge of G is contained in a clique of order greater than or equal to 4. Tuza (1990) asked if tau(c)(G) less than or equal to \G\/4 for all G E G, where \G\ denotes the number of vertices of G. Flotow (1992) had constructed infinitely many examples each showing that the answer to this question is negative. In the present note, by modifying these examples, we show that, for each epsilon > 0, there exists a graph G is an element of such that tau(c)(G) greater than or equal to (2/7 - epsilon)\G\. We conjecture tau(c)(G)less than or equal to 2\G\7 for all G is an element of G and present a partial result supporting this conjecture.