arXiv subjects

Hyesun Yoon

Publications and source records attributed to Hyesun Yoon.

4 recordsLinked to original sources

arXiv2024

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

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.

math.CO↗

arXiv2022

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

For an integer $k \ge 2$, let $A$ be a Boolean block matrix with blocks $A_{ij}$ for $1 \le i,j \le k$ such that $A_{ii}$ is a zero matrix and $A_{ij}+A_{ji}^T$ is a matrix with all elements $1$ but not both corresponding elements of $A_{ij}$ and $A_{ji}^T$ equal to $1$ for $i \neq j$. Jung~{\em et al.} [Competition periods of multipartite tournaments. {\it Linear and Multilinear Algebra}, https://doi.org/10.1080/03081087.2022.2038057] studied the matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$. This paper, which is a natural extension of the above paper and was initiated by the observation that $\{A^m(A^T)^m\}_{m=1}^{\infty}$ converges if $A$ has no zero rows, computes the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$ if $A$ has no zero rows. To this end, we take a graph theoretical approach: noting that $A$ is the adjacency matrix of a multipartite tournament $D$, we compute the limit of the graph sequence $\left\{ C^m(D) \right\}_{m=1}^{\infty}$ when $D$ has no sinks.

math.CO↗

arXiv2020

On competition indices and periods of multipartite tournaments

In this paper, we compute competition indices and periods of multipartite tournaments. We first show that the competition period of an acyclic digraph $D$ is one and $ζ(D) +1$ is a sharp upper bound of the competition index of $D$ where $ζ(D)$ is the sink elimination index of $D$. Then we prove that, especially, for an acyclic $k$-partite tournament $D$, the competition index of $D$ is $ζ(D)$ or $ζ(D) +1$ for an integer $k \ge 3$. By developing useful tools to create infinitely many directed walks in a certain regular pattern from given directed walks, we show that the competition period of a multipartite tournament with sinks and directed cycles is at most three. We also prove that the competition index of a primitive digraph does not exceed its exponent.

math.CO↗

arXiv2019

On $m$-step competition graphs of bipartite tournaments

In this paper, we completely characterize the $m$-step competition graph of a bipartite tournament for any integer $m \ge 2$. In addition, we compute the competition index and the competition period of a bipartite tournament.

math.CO↗