On optimal orientations of complete tripartite graphs

Abstract

Given a connected and bridgeless graph <i>G</i>, let <i>D</i>(<i>G</i>) be the family of strong orientations of <i>G</i>. The orientation number of <i>G</i> is defined to be đ(<i>G</i>) := min{<i>d</i>(<i>D</i>) | <i>D</i> ∈ <i>D</i>(<i>G</i>)}, where ,<i>d</i>(<i>D</i>) is the diameter of the digraph <i>D</i>. In this paper, we focus on the orientation number of complete tripartite graphs. We prove a conjecture raised by Rajasekaran and Sampathkumar. Specifically, for q ≥ p ≥ 3, if đ(<i>K</i>(2, <i>p, q</i>)) = 2, then q ≤ (^p_└p/2┘). We also present some sufficient conditions on <i>p</i> and <i>q</i> for đ(<i>K</i>(<i>p, p, q</i>)) = 2.