The interval number of a complete multipartite graph

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 n1≥n2≥ ⋯ ≥np. West showed that for each n ≥ 3, there exists a constant cn so that if p ≥ cn,n1 = n2−n −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 n1 ≥ n2 so that i(K(n2,…,np)) = i(K(n1,n2)) whenever p ≥ 2 and n2 ≥ n3 ≥ ⋯ ≥ np. In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n2 − n − 1, n) for n ≥ 3 and (7,5).