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 $f_d(n)$ denotes the minimum number of arcs in an oriented graph on $n$ vertices having weak diameter $d$. Finding the exact value of $f_d(n)$ is a challenging problem even for $d = 2$. This function was introduced by Katona and Szemeredi (1967), and after that several attempts were made to find its exact value by Znam (1970), Dawes and Meijer (1987), Furedi, 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 behaviour 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 $f_2(n)$ is a strictly increasing function. Furthermore, we improve the best known upper bound of $f_2(n)$ and conjecture that our upper bound is tight. This improvement of the upper bound improves known bounds involving the oriented achromatic number.

翻译：一个定向图具有弱直径至多为$d$，如果每对不相邻顶点均由一条有向$d$路径连接。函数$f_d(n)$表示在$n$个顶点上具有弱直径$d$的定向图所需的最小弧数。即使对于$d = 2$，精确确定$f_d(n)$的值也是一个具有挑战性的问题。该函数由Katona和Szemerédi于1967年引入，此后Znam（1970）、Dawes和Meijer（1987）、Füredi、Horak、Pareek和Zhu（1998），以及Kostochka、Łuczak、Simonyi和Sopena（1999）通过改进其最佳已知界试图确定其精确值。在此过程中，证明了该函数渐近等于$n\log_2 n$，因此是一个渐近递增函数。然而，该函数的精确取值和行为此前尚不明确。本文发现，弱直径至多为$2$的定向图恰好是绝对定向团，即定向着色背景下定向图相对于团的类比。通过研究弧最小绝对定向团，我们证明$f_2(n)$是一个严格递增函数。此外，我们改进了$f_2(n)$的最佳已知上界，并推测该上界是紧的。该上界的改进提升了涉及定向无彩色数的已知界。