The minimum diameter of orientations of complete multipartite graphs

Abstract

Given a graph G, let [Figure not available: see fulltext.], and define [Figure not available: see fulltext.] A pair {p,q} of integers is called a co-pair if $$1 \leqslant p \leqslant q \leqslant \left( {\begin{array}{*{20}c} p \\ {\left\lfloor {\frac{p} {2}} \right\rfloor } \\ \end{array} } \right)$$ . A multiset {p, q, r} of positive integers is called a co-triple if {p, q} and {p, r} are co-pairs. Let K(p 1, p 2,..., p n ) denote the complete n-partite graph havingp i vertices in the ith partite set. In this paper, we show that if {p 1, p 2,...,p n } can be partitioned into co-pairs whenn is even, and into co-pairs and a co-triple whenn is odd, then?(K(p 1, p 2,..., p n )) = 2 provided that (n,p 1, p 2, p 3, p 4) ? (4, 1, 1, 1, 1). This substantially extends a result of Gutin [3] and a result of Koh and Tan [4].