The interval number of a complete multipartite graph

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Abstract

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t closed intervals of the real line R. Trotter and Harary showed that the interval number of the complete bipartite graph K(m,n) is (mn + 1)(m + n). Matthews showed that the interval number of the complete multipartite graph K(n1,n2,…,np) was the same as the interval number of K(n1,n2) when n1 = n2 = ⋯ = np. Trotter and Hopkins showed that i(K(n1,n2,…,np)) ≤ 1 + i(K(n1,n2)) whenever p ≥ 2 and n1n2≥ ⋯ ≥np. West showed that for each n ≥ 3, there exists a constant cn so that if pcn,n1 = n2n −1, and n2 = n3 = ⋯ np = n, then i(K(n1,n2,…,np) = 1 + i(K(n1, n2)). In view of these results, it is natural to consider the problem of determining those pairs (n1,n2) with n1n2 so that i(K(n2,…,np)) = i(K(n1,n2)) whenever p ≥ 2 and n2n3 ≥ ⋯ ≥ np. In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n2n − 1, n) for n ≥ 3 and (7,5).

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Research supported in part by NSF Grants ISP-8011451 and MCS-8202172.