Counting the minimum number of arcs in an oriented graph having weak diameter 2

Abstract

An oriented graph has weak diameter at most $d$ if every non-adjacent pair of vertices are connected by a directed $d$-path. The function $f_d\left(n\right)$ denotes the minimum number of arcs in an oriented graph on $n$ vertices having weak diameter $d$. Finding the exact value of $f_d\left(n\right)$ is a challenging problem even for $d=2$. This function was introduced by Katona and Szemeŕedi (1967), and after that several attempts were made to find its exact value by Znam (1970), Dawes and Meijer (1987), Füredi, Horak, Pareek and Zhu (1998), and Kostochka, Luczak, Simonyi and Sopena (1999) through improving its best known bounds. In that process, it was proved that this function is asymptotically equal to $n{log}_{2}n$ and hence, is an asymptotically increasing function. However, the exact value and behavior of this function was not known.

In this article, we observe that the oriented graphs with weak diameter at most 2 are precisely the absolute oriented cliques, that is, analogues of cliques for oriented graphs in the context of oriented coloring. Through studying arc-minimal absolute oriented cliques we prove that $f_2\left(n\right)$ is a strictly increasing function. Furthermore, we improve the best known upper bound of $f_2\left(n\right)$ and conjecture that our upper bound is tight. This improvement of the upper bound improves known bounds involving the oriented achromatic number.