On the Generalized Competition Index of a Regular or Almost Regular Tournament

Abstract

For a primitive digraph \(D\) of order \(n\) and a positive integer \(m\) such that \(1 \leq m \leq n\), we define the \(m\)-competition index of \(D\), denoted by \(k_m(D)\), as the smallest positive integer \(k\) such that distinct vertices \(v_1, v_2, \ldots, v_m\) exist for each pair of vertices \(x\) and \(y\) with \(x \rightarrow^k v_i\) and \(y \rightarrow^k v_i\) for \(1 \leq i \leq m\) in \(D\). In this paper, we investigate the \(m\)-competition index of regular or almost regular tournaments.