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On covering all cliques of a chordal graph. (English) Zbl 0846.05050
Let \(G\) be a graph without isolated vertices. A set \(T\subseteq V(G)\) is called a clique-transversal set if \(T\) has non-empty intersection with every maximal clique in \(G\). The clique-transversal number \(\tau_c(G)\) is the minimum cardinality of a clique-transversal set. Let \(\mathcal G\) denote the class of chordal graphs with the property that each edge is contained in a clique of order at least 4. The authors show that for each \(\varepsilon> 0\), there exists a graph \(G\in {\mathcal G}\) such that \(\tau_c(G)\geq ({2\over 7}- \varepsilon)\). They give some evidence to support their conjecture that \(\tau_c(G)\leq {2\over 7} |V(G)|\) for all \(G\in {\mathcal G}\).

05C35 Extremal problems in graph theory
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
Full Text: DOI
[1] Andreae, T.; Schughart, M.; Tuza, Zs., Clique-transversal sets of line graphs and complements of line graphs, Discrete math., 88, 11-20, (1991) · Zbl 0734.05077
[2] Duffus, D.; Kierstead, H.A.; Trotter, W.T., Fibres and ordered set coloring, J. combin. theory ser. A, 58, 158-164, (1991) · Zbl 0757.06001
[3] Duffus, D.; Sands, B.; Sauer, N.; Woodrow, R.E., Two-coloring all two-element maximal antichains, J. combin. theory ser. A, 57, 109-116, (1991) · Zbl 0742.06004
[4] Erdős, P.; Gallai, T.; Tuza, Zs., Covering the cliques of a graph with vertices, Discrete math., 108, 279-289, (1992) · Zbl 0766.05063
[5] Flotow, C., Obere schranken fuer die clique-transversalzahl eines graphen, ()
[6] Lonc, Z.; Rival, I., Chains, antichains and fibres, J. combin. theory ser. A, 44, 207-228, (1987) · Zbl 0637.06001
[7] Maltby, R., A smallest-fibre-size to poset-size ratio approaching \(815\), J. combin. theory ser. A, 61, 328-330, (1992) · Zbl 0766.06006
[8] Tuza, Zs., Covering all cliques of a graph, Discrete math., 86, 117-126, (1990) · Zbl 0744.05040
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