By Jagdish Srivastava
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Additional resources for A Survey of Combinatorial Theory
1) => (2), because, for any hypergraph H9 we have 2v(i7) < τ2(Η) < 2τ(Η). Let us show that (2) => (1). Let us assume that a hypergraph H satisfies (2) and v(H) < τ(Η)9 and let us show that this leads to a contradiction. Let //' be a partial hypergraph of H with v(H') < τ(Η') and with a minimal number of edges. For every edge E[ of //', we have ν(Η'-Εί) = τ(Η'-Ε& v(//'-ED = v(H') or v(H')-l9 τ(Η'-Ε[) = τ(Η') or τ(Η')-1. Hence ν(Η'-£ί) = ν(Η') (0, τ(Α'-£ί) = τ(/Τ)-1 (0· , Let £Ί and JE2 be two intersecting edges of H' (they exist by the above equalities).
Proof. Let (Sl9 S29 · . ·, Sk) be a partition of X into k classes, and let k(i) be the number of classes which meet edge Et. If k(i) = k for every /, then we have a partition of X into k transversal sets. If k(i) < k for an index i = i09 we have k(i0) < \Eio\; hence there exist two indices p and q with \SP n JEiol > 2, |S e n £ fe | = 0. The sub-hypergraph / ί ' induced by 5 Ρ u Sq is balanced, hence, by theorem 1, it admits a bicoloration (S'p, S£). Let Sj = Sj for j Φ p,q\ the partition (Si, S £ , .
T. ; Academic Press, New York), pp. 49-57. C. Berge, 1970, Graphes et Hypergraphes (Dunod, Paris), ch. 20. C. Berge and M. Las Vergnas, 1970, Sur un theoreme du type König pour hypergraphes, Intern. Conf. on Combinatorial Mathematics (A. Gerwitz, L. ), Ann. New York Acad. Sei. 175, 32-40. P. Erdös and A. Hajnal, 1966, On chromatic number of graphs and set-systems, ActaMath. Acad. Sei. Hungarica 17, 61-99. D. R. Fulkerson, 1970, Anti-blocking polyhedra, Rand Rept. No. RM-6201. P. M. Res. C. 184.
A Survey of Combinatorial Theory by Jagdish Srivastava