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Covering all cliques of a graph. (English) Zbl 0744.05040
Summary: The following conjecture of T. Gallai is proved: If \(G\) is a chordal graph on \(n\) vertices, such that all its maximal complete subgraphs have order at least 3, then there is a vertex set of cardinality \(\leq n/3\) which meets all maximal complete subgraphs of \(G\). Further related results are given.

MSC:
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C35 Extremal problems in graph theory
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[1] Aigner, M.; Andreae, T., Vertex-sets that meet all maximal cliques of a graph, (1986), manuscript
[2] Andreae, T., Covering the maximal cliques of a graph with at most half of its vertices, (1987), manuscript
[3] Berge, C., Graphs and hypergraphs, (1973), North-Holland Amsterdam · Zbl 0483.05029
[4] Dirac, G.A., On rigid circuit graphs, Abh. math. sem. univ. Hamburg, 25, 71-76, (1961) · Zbl 0098.14703
[5] P. Erdős, T. Gallai and Zs. Tuza, Covering the cliques of a graph with vertices, Discrete Math., to appear.
[6] Farber, M., Characterizations of strongly chordal graphs, Discrete math., 43, 173-189, (1983) · Zbl 0514.05048
[7] Gallai, T., Über extreme punkt- und kantenmengen, Ann. univ. sci. L. Eötvös sect. math., 2, 133-138, (1959) · Zbl 0094.36105
[8] Tuza, Zs., Extremal problems on graphs and hypergraphs, () · Zbl 0584.05042
[9] Zs. Tuza, Matchings and coverings in regular uniform hypergraphs, to appear. · Zbl 0541.05046
[10] Alon, N., Transversal numbers of uniform hypergraphs, Graphs and combinatorics, 6, 1-4, (1990) · Zbl 0742.05065
[11] T. Andreae, M. Schughart, and Zs. Tuza, Clique-transversal sets of line graphs and complements of line graphs, Discrete Math., to appear. · Zbl 0734.05077
[12] Feng-Chu Lai and G.J. Chang, An upper bound for the transversal numbers of 4-uniform hypergraphs, J. Combin. Theory Ser. B, to appear. · Zbl 0739.05068
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