Counting the minimum number of arcs in an oriented graph having weak diameter 2
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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 Szemeŕedi [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], Füredi, 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 nlog_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.
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