A t-(v, k, A)-design, say (X,A), is said to be a graphical t-design if X consists...

 

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A t-(v, k, A)-design, say (X,A), is said to be a graphical t-design if X consists of the edges of a complete graph K, (i.e., all the 2-subsets of a v-set, where v = ()) and A is formed by taking all subgraphs of Kn that are isomorphic to one of the graphs in a set 9 of specified sub- graphs of K. (a) Suppose that v = 15, n = 6, and G= {{12,34,56}, {12, 13, 23}}. Prove that the result is a graphical 2-(15,3, 1)-design (i.e., a (15,3,1)-BIBD). (b) Suppose that v = 10, n = 5, and G = {{12, 13, 14, 15}, {12, 13, 23, 45}, {12, 23, 34, 14}}. Prove that the result is a graphical 3-(10,4, 1)-design. A t-(v, k, A)-design, say (X,A), is said to be a graphical t-design if X consists of the edges of a complete graph K, (i.e., all the 2-subsets of a v-set, where v = ()) and A is formed by taking all subgraphs of Kn that are isomorphic to one of the graphs in a set 9 of specified sub- graphs of K. (a) Suppose that v = 15, n = 6, and G= {{12,34,56}, {12, 13, 23}}. Prove that the result is a graphical 2-(15,3, 1)-design (i.e., a (15,3,1)-BIBD). (b) Suppose that v = 10, n = 5, and G = {{12, 13, 14, 15}, {12, 13, 23, 45}, {12, 23, 34, 14}}. Prove that the result is a graphical 3-(10,4, 1)-design.