The interval number of a complete multipartite graph

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 cIosed intervals of the real line IR. Trotter and Harary showed that the interval number of the complete bipartite graph K(m, n) is [(mn + I)/(m + n)]. Matthews showed that the interval number of the complete multipartite graph K(n,,n2, . . ..np) was the same as the interval number of K(n,,Q when n, = n2 = ... =np. Trotter and Hopkins showed that i(K(n,, ttr, . . ,n,,))s I + i(K(n,,nl)) whenever pr2 and n,~n~z...zn~. West showed that for each nr3, there exists a constant c, so that if pzc,, n, =n’n-1,andnz=n?=...n,=n,theni(k’(n,,n:,...,n,))=l+i(K(n,,n,)). Inviewof these results, it is natural to consider the problem of determining those pairs (n,, n.) with n, >nL so that i(K(nz,....nP))=i(K(n,,nz)) whenever p?2 and n2tn3z...znp, In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n’ n I, n) for n e 3 and (7,5).