Graphs and Combinatorics (1994) 10:225-230 Graphs and Combinatorics 9 Springer-Verlag 1994 Minimizing and Maximizing the Diameter in Orientations of Graphs G. Gutin Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv 69978, Israel Abstract. For a graph G, let G'(G") denote an orientation of G having maximum (mini- mum respectively) finite diameter. We show that the length of the longest path in any 2-edge connected (undirected) graph G is precisely diam(G'). Let K(mt,m 2..... m,) be the complete n-partite graph with parts of cardinalities m l, m2 ..... m~. We prove that if m 1 = m 2 .... = m. = m, n >_ 3, then diam(K"(ml,m 2 ..... m.)) = 2, unless m = 1 and n=4. 1. Introduction The following is a well known Theorem of Robbings [1]; a connected graph G has a strongly connected orientation if and only if G has no bridge. Therefore, we consider here only (connected) graphs without bridges (an edge e of a (connected) graph G is called a bridge if G - e is not connected). For a graph G, let G'(G") denote an orientation of G having maximum (minimum, respectively) finite diameter. In this work we prove that for any graph G diam(G') is equal to the length of the longest path of G (denoting here by Ip(G)). This implies the inequality of Ghouila-Houri (cf. [2], page 72) for oriented graphs. Define f(ml, m 2 .... , mn) = diam(K"(ml, m 2 ..... mn) ). Boesh and Tindell I-3] proved that f(m,m)=3 for m>2. Plesnik 1-4] showed that if m i, m 2>2, then f(ml, m2) < 4. Finally, Soltes [4] determined the exact value of f(m 1, m2) for ( m2 ),andotherwise all ml, m 2. Ifmi _> m 2 _> 2, then f(mi,m2) = 3 for m i < Lm2/2j f(ml, m2) = 4. A short proof of this result, using the well known theorem of Sperner is given in [5]. In the present paper we prove that if n >_ 3, then f(rn~, m 2 ..... ran) _ 3 for all m~ (i = 1, 2 ..... n) and determine f(ml ..... ran) precisely for all m i = trt2 ..... rn n = m; ifn _> 3 thenf(mi,m 2 ..... m.) = 2 unless n = 4 and m = 1.