Maximum oriented forcing number for complete graphs

TL;DR

This paper investigates the maximum oriented forcing number of complete graphs, providing bounds and exploring its implications for tournaments and related graph classes, especially focusing on the case when k=1.

Contribution

It establishes new lower bounds for the maximum oriented forcing number of complete graphs and extends the analysis to complete q-partite graphs, advancing understanding of this invariant.

Findings

Lower bound of roughly 3/4 n for F(K_n)

Lower bound of n - 2n / log_2(n) for n  2

Analysis of F for complete q-partite graphs

Abstract

The maximum oriented $k$-forcing number of a simple graph $G$, written $\MOF_k(G)$, is the maximum directed $k$-forcing number among all orientations of $G$. This invariant was recently introduced by Caro, Davila and Pepper in [CaroDavilaPepper], and in the current paper we study the special case where $G$ is the complete graph with order $n$, denoted $K_n$. While $\MOF_k(G)$ is an invariant for the underlying simple graph $G$, $\MOF_k(K_n)$ can also be interpreted as an interesting property for tournaments. Our main results further focus on the case when $k=1$. These include a lower bound on $\MOF(K_n)$ of roughly $\frac{3}{4}n$, and for $n\ge 2$, a lower bound of $n - \frac{2n}{\log_2(n)}$. Along the way, we also consider various lower bounds on the maximum oriented $k$-forcing number for the closely related complete $q$-partite graphs.