The minimum diameter of orientations of complete multipartite graphs

Abstract

Given a graph G, let

, and define

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, thenε(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].