On the convergence of the graph sequence $\left\{ C^m(D) \right\}_{m=1}^{\infty}$ for a multipartite tournament $D$

Abstract

Given a positive integer $m$, the $m$-step competition graph of a digraph $D$, denoted by $C^m(D)$, has the same vertex set as $D$ and has an edge between vertices $u$ and $v$ if and only if there exists a vertex $w$ such that there exist directed walks of length $m$ from $u$ to $w$ and from $v$ to $w$, respectively. In this paper, we completely characterize the convergence of $\{C^m(D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$ based on the last nontrivial strong component of $D$. Furthermore, not only do we determine the limit in the case of convergence, but also in the event of divergence, we specify how $C^m(D)$ changes periodically depending on the value of $m$. Our results extend the work of Jung et al. [On the limit of the sequence $\{C^m (D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$. Discrete Appl. Math., 340:1--13, 2023] which addresses the case of the last strong component being nontrivial, thereby completing the convergence analysis of $\{C^m(D)\}_{m=1}^{\infty}$ for a multipartite tournament $D$. Our results can also be expressed in terms of matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$ for the adjacency matrix $A$ of $D$ and this part is also covered in the text.

Figures (6)

• Figure 1: A tripartite tournament $D$ and its adjacency matrix $A$ where $V_1 =\{v_1\}$, $V_2 =\{v_2,v_3,v_4\}$, and $V_3=\{v_5, v_6\}$ are the partite sets of $D$.
• Figure 2: Solid-line boxes represent the converging cases, while dashed-line boxes represent the diverging cases.
• Figure 3: Tripartite tournaments $D_1$ and $D_2$ and their ordered strong components $Q_1, Q_2, Q_3$.
• Figure 4: The graphs $G_1$, $G_2$, and $G_3$ in Theorem \ref{['thm:last']} (refer to \ref{['figure']} for other notations).
• Figure 5: The graphs $G_4$ and $G_5$ in Theorem \ref{['thm:4']} where $K^{(1)} = K\left[D_{1 \sim r}\right]$, $K^{(2)} = K\left[A_1 - V(Q_t) \right]$, $K^{(3)} = K\left[A_2 - V(Q_t) \right]$, $K^{(4)} = K\left[U_{1}^{(t)}\right]$, $K^{(5)} = K\left[U_{3}^{(t)}\right]$, $K^{(6)} = K\left[U_{2}^{(t)}\right]$, and $K^{(7)} = K\left[U_{4}^{(t)}\right]$ (refer to \ref{['figure']} for other notations). Here, $r$ is the head completing index of $D$ and $A_1$ and $A_2$ are given in \ref{['a12']}.

Theorems & Definitions (18)

• Proposition 1.1: jung2023limit
• Theorem 1.2
• Example 1.3
• Theorem 2.1: jung2023limit
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 2.5
• proof
• Proposition 2.6: jung2023limit