By Jagdish Srivastava

ISBN-10: 0720422620

ISBN-13: 9780720422627

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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 £ , .

The three following statements are equivalent: (1) OL(GA,D) = 9(GA,D) for every A and every D. (2) y(GA,D) = CO(GA,D) for every A and every D. (3) every odd cycle in G contains at least one edge with the property that every maximal clique containing this edge contains a third vertex of the cycle. If we denote by H = (Z, C) the hypergraph of the maximal cliques of G, (3) is equivalent to (3') / / i s balanced, CH. 2 BALANCED HYPERGRAPHS AND APPLICATIONS TO GRAPH THEORY 23 or to (3") the dual H* of H is balanced.

3 discovered by Golay [1949]. It was discussed by Coxeter [1958] in a geometric context, and by Bose [1961], who indicated its connection to the theory of confounding and fractional replication. Bridges and Ryser [1969] considered yet another generalization of block designs. Their («, k, 2)-systems on r and s are defined in terms of binary square matrices X and Y of order n satisfying the real matrix equation! XY = γχ= (k-X)I+XJ, k*k9 which, for λ Φ 0, implies JX = XJ = rJ, JY = YJ = sJ for integer r and s.

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A Survey of Combinatorial Theory by Jagdish Srivastava

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