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

TL;DR

This work investigates the minimum arc count function $f_2(n)$ for oriented graphs with weak diameter $2$, showing that such graphs are precisely the absolute oriented cliques and establishing that $f_2(n)$ is strictly increasing in $n$. It introduces a recursive sequence $x_n$ with initial values $x_1=0, x_2=1, x_3=2, x_4=4, x_5=5$ and a case-based recurrence, proving that $f_2(n)\le x_n$ and conjecturing $f_2(n)=x_n$ for all $n$. A constructive upper bound is derived via a family of absolute oriented cliques $\overrightarrow{O}_n$ built with a semidirect product operation $\ltimes$, achieving $|A(\overrightarrow{O}_n)|=x_n$; this tightens previous bounds on the oriented achromatic number. The results have implications for oriented coloring theory and suggest future extensions to colored mixed graphs, while providing concrete structural insights into extremal absolute oriented cliques.

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 $f_d(n)$ 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.

Figures (3)

• Figure 1: The absolute oriented cliques $\overrightarrow{O}_1, \overrightarrow{O}_2, \overrightarrow{O}_3, \overrightarrow{O}_4$, and $\overrightarrow{O}_5$.
• Figure 2: The oriented graph $\overrightarrow{G}_1 \ltimes \overrightarrow{G}_2$.
• Figure 3: Comparision of $f_2(n)$ upper bounds as obtained by the various results mentioned in this paper

Theorems & Definitions (41)

• Theorem 1.1: Klostermeyer and MacGillivray klostermeyer2004analogues
• Remark 1.2
• Theorem 1.3: Katona and Szemerédi katona1967problem
• Theorem 1.4: Füredi, Horak, Pareek and Zhu furedi1998minimal
• Theorem 1.5: Kostochka, Luczak, Simonyi and Sopena kostochka1999minimum
• Theorem 1.6: Füredi, Horak, Pareek and Zhu furedi1998minimal and Kostochka, Luczak, Simonyi and Sopena kostochka1999minimum
• Theorem 1.7
• Definition 1.8
• Conjecture 1.9
• Theorem 1.10