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 $d$. It turns out that finding the exact value of $f_d(n)$ is a challenging problem even for $d = 2$. This function was introduced by Katona and Szeme\'redi [Studia Scientiarum Mathematicarum Hungarica, 1967], and after that several attempts were made to find its exact value by Znam [Acta Fac. Rerum Natur. Univ. Comenian. Math. Publ, 1970], Dawes and Meijer [J. Combin. Math. and Combin. Comput, 1987], F\"uredi, Horak, Pareek and Zhu [Graphs and Combinatorics, 1998], and Kostochka, Luczak, Simonyi and Sopena [Discrete Mathematics and Theoretical Computer Science, 1999] through improving its best known upper bounds. In that process, they also proved that this function is asymptotically equal to $n\log_2 n$ and hence, is an asymptotically increasing function. In this article, we prove that $f(n)$ is a strictly increasing function. Furthermore, we improve the best known upper bound of $f(n)$ and conjecture that it is tight.

翻译：一个定向图具有弱直径至多$d$，如果每对不相邻的顶点都通过一条有向$d$-路径相连。函数$f_d(n)$表示$n$个顶点上具有弱直径$d$的有向图所需的最少弧数。即使对于$d=2$的情况，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 [Studia Scientiarum Mathematicarum Hungarica, 1967]提出，此后Znam [Acta Fac. Rerum Natur. Univ. Comenian. Math. Publ, 1970]、Dawes和Meijer [J. Combin. Math. and Combin. Comput, 1987]、Füredi、Horak、Pareek和Zhu [Graphs and Combinatorics, 1998]以及Kostochka、Luczak、Simonyi和Sopena [Discrete Mathematics and Theoretical Computer Science, 1999]通过改进其已知最优上界，多次尝试求解其精确值。在此过程中，他们还证明该函数渐近等于$n\log_2 n$，因此是一个渐近递增函数。本文中，我们证明了$f(n)$是严格递增函数。此外，我们改进了$f(n)$的已知最优上界，并猜想该上界是紧的。