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Suh-Ryung Kim

Publications and source records attributed to Suh-Ryung Kim.

At least 19 recordsLinked to original sources
arXiv2025

A digraph version of the Friendship Theorem

The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erdős, Alfréd Rényi, and Vera T. Sós in 1966. ``What would happen if instead any pair of persons likes precisely one person?" While a friendship relation is symmetric, a liking relation may not be symmetric. Therefore to represent a liking relation we should use a directed graph. We call this digraph a ``liking digraph". It is easy to check that a symmetric liking digraph becomes a friendship graph if each directed cycle of length two is replaced with an edge. In this paper, we provide a digraph formulation of the Friendship Theorem which characterizes the liking digraphs. We also establish a sufficient and necessary condition for the existence of liking digraphs.

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arXiv2025

Competition graphs of degree bounded digraphs

If each vertex of an acyclic digraph has indegree at most $i$ and outdegree at most $j$, then it is called an $(i,j)$ digraph, which was introduced by Hefner~{\it et al.}~(1991). Whereas Hefner~{\it et al.} characterized $(i,j)$ digraphs whose competition graphs are interval, characterizing the competition graphs of $(i,j)$ digraphs is not an easy task. In this paper, we introduce the concept of $\langle i,j \rangle$ digraphs, which relax the acyclicity condition of $(i,j)$ digraphs, and study their competition graphs. By doing so, we obtain quite meaningful results. Firstly, we give a necessary and sufficient condition for a loopless graph being an $\langle i,j \rangle$ competition graph for some positive integers $i$ and $j$. Then we study on an $\langle i,j \rangle$ competition graph being chordal and present a forbidden subdigraph characterization. Finally, we study the family of $\langle i,j \rangle$ competition graphs, denoted by $\mathcal{G}_{\langle i,j \rangle}$, and identify the set containment relation on $\{\mathcal{G}_{\langle i,j \rangle}\colon\, i,j \ge 1\}$.

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arXiv2025

Digraphs in which every $t$ vertices have exactly $λ$ common out-neighbors

We say that a digraph is a $(t,λ)$-liking digraph if every $t$ vertices have exactly $λ$ common out-neighbors. In 1975, Plesník [Graphs with a homogeneity, 1975. {\it Glasnik Mathematicki} 10:9-23] proved that any $(t,1)$-liking digraph is the complete digraph on $t+1$ vertices for each $t\geq 3$. Choi {\it et al}. [A digraph version of the Friendship Theorem, 2025. {\it Discrete mathematics}, 348(1), 114238] showed that a $(2,1)$-liking digraph is a fancy wheel digraph or a $k$-diregular digraph for some positive integer $k$. In this paper, we extend these results by completely characterizing the $(t,λ)$-liking digraphs with $t \geq λ+2$ and giving some equivalent conditions for a $(t,λ)$-liking digraph being a complete digraph on $t+λ$ vertices.

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arXiv2024

Extensions of results on phylogeny graphs of degree bounded digraphs

An acyclic digraph in which every vertex has indegree at most $i$ and outdegree at most $j$ is called an $(i,j)$ digraph for some positive integers $i$ and $j$. The phylogeny graph of a digraph $D$ has $V(D)$ as the vertex set and an edge $uv$ if and only if one of the following is true: $(u,v) \in A(D)$; $(v,u) \in A(D)$; $(u,w) \in A(D)$ and $(v,w) \in A(D)$ for some $w \in V(D)$. A graph $G$ is a phylogeny graph (resp.\ an $(i,j)$ phylogeny graph) if there is an acyclic digraph $D$ (resp.\ an $(i,j)$ digraph $D$) such that the phylogeny graph of $D$ is isomorphic to $G$. Lee~{\em et al.} (2017) and Eoh and Kim (2021) studied the $(2,2)$ phylogeny graphs, $(1,j)$ phylogeny graphs, $(i,1)$ phylogeny graphs, and $(2,j)$ phylogeny graphs. Their work was motivated by problems related to evidence propagation in a Bayesian network for which it is useful to know which acyclic digraphs have chordal moral graphs (phylogeny graphs are called moral graphs in Bayesian network theory). In this paper, we extend their work by giving necessary conditions of chordal $(i,2)$ phylogeny graphs. We go further to give necessary conditions of $(i,j)$ phylogeny graphs by listing forbidden induced subgraphs.

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arXiv2024

Multipartite tournaments in which any two vertices have an $(i,j)$-step common out-neighbor

We say that a digraph $D$ is $(i,j)$-step competitive if any two vertices have an $(i,j)$-step common out-neighbor in $D$ and that a graph $G$ is $(i,j)$-step competitively orientable if there exists an $(i,j)$-step competitive orientation of $G$. In [Choi et al. Competitively orientable complete multipartite graphs. Discrete Mathematics, 345(9):112950, 2022], Choi et al. introduce the notion of competitive digraph and completely characterize competitively orientable complete multipartite graphs in terms of the sizes of its partite sets. Here, a competitive digraph means a $(1,1)$-step competitive digraph. In this paper, the result of Choi et al. has been extended to a general characterization of $(i,j)$-step competitively orientable complete multipartite graphs.

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arXiv2024

On Toeplitz graphs being line graphs

A Toeplitz graph $T_n \langle t_1,t_2,\ldots,t_k\rangle$ is a simple graph with the vertex set $[n]$ such that two vertices $v$ and $w$ are adjacent if and only if $|v-w| = t_i$ for some $i \in [k]$. In this paper, we investigate line Toeplitz graphs, which are Toeplitz graphs that happen to be line graphs. We first show that for a sufficiently large $n$, the family of claw-free Toeplitz graphs of order $n$ is $T_n \langle t,2t,\ldots,kt\rangle$ for some nonnegative integers $t$ and $k$. Interestingly, this family consists of a union of Toeplitz graphs each of which is isomorphic to a $k$-tree the notion of which was introduced by Patil in 1986. Then we completely characterize $T_n \langle t,2t,\ldots,kt\rangle$ for any positive integer $n$ that is a line graph. Furthermore, we provide a comprehensive description of a line Toeplitz graph $T_n \langle t_1,t_2\rangle$ and $T_n \langle t_1,t_2,t_3\rangle$. In general, line Toeplitz graph seems very challenging to characterize completely. Even for $T_n \langle t_1,t_2,t_3\rangle$, it was not easy to do so. It is also worth mentioning that there is a line Toeplitz graph that is not in the form $T_n \langle t,2t,3t\rangle$.

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arXiv2024

Matrix periods and competition periods of Boolean Toeplitz matrices II

This paper is a follow-up to the paper [Matrix periods and competition periods of Boolean Toeplitz matrices, {\it Linear Algebra Appl.} 672:228--250, (2023)]. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices $A=T_n\langle S; T \rangle$ satisfying the condition ($\star$) $\max S+\min T \le n$ and $\min S+\max T \le n$ are $d^+/d$ and $1$, respectively, where $d^+= \gcd (s+t \mid s \in S, t \in T)$ and $d = \gcd(d, \min S)$. In this paper, we claim that even if ($\star$) is relaxed to the existence of elements $s \in S$ and $t \in T$ satisfying $s+t \le n$ and $\gcd(s,t)=1$, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy ($\star$) but the relaxed condition. For example, for any positive integers $k, n$ with $2k+1 \le n$, it is easy to see that $T_n\langle k, n-k;k+1, n-k-1 \rangle$ does not satisfies ($\star$) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is $T_n\langle d^+,2d^+, \ldots, \lfloor n/d^+\rfloor d^+\rangle$.

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arXiv2024

Digraphs in which every $t$ vertices share exactly $λ$ out-neighbors and exactly $λ$ in-neighbors

In this paper, we introduce the notion of two-way $(t,λ)$-liking digraphs as a way to extend the results for generalized friendship graphs. A two-way $(t,λ)$-liking digraph is a digraph in which every $t$ vertices have exactly $λ$ common out-neighbors and $λ$ common in-neighbors. We first show that if $λ\ge 2$, then a two-way $(2,λ)$-liking digraph of order $n$ is $k$-diregular for a positive integer $k$ satisfying the equation $(n-1)λ=k(k-1)$. This result is comparable to the result by Bose and Shrikhande in 1969 and actually extends it. Another main result is that if $t \ge 3$, then the complete digraph on $t+λ$ vertices is the only two-way $(t,λ)$-liking digraph. This result can stand up to the result by Carstens and Kruse in 1977 and essentially extends it. In addition, we find that two-way $(t, λ)$-liking digraphs are closely linked to symmetric block designs and extend some existing results of $(t, λ)$-liking digraphs.

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arXiv2024

Competition-common enemy graphs of degree-bounded digraphs

The competition-common enemy graph (CCE graph) of a digraph $D$ is the graph with the vertex set $V(D)$ and an edge $uv$ if and only if $u$ and $v$ have a common predator and a common prey in $D$. If each vertex of a digraph $D$ has indegree at most $i$ and outdegree at most $j$, then $D$ is called an $\langle i,j \rangle$ digraph. In this paper, we fully characterize the CCE graphs of $\langle 2,2\rangle$ digraphs. Then we investigate the CCE graphs of acyclic $\langle 2,2 \rangle$ digraphs, and prove that any CCE graph of an acyclic $\langle 2,2 \rangle$ digraph with at most seven components is interval, and the bound is sharp. While characterizing acyclic $\langle 2,2 \rangle$ digraphs that have interval graphs as their competition graphs, Hefner~{\it et al}. (1991) initiated the study of competition graphs of degree-bounded digraphs. Recently, Lee~{\em et al}. (2017) and Eoh and Kim (2021) studied phylogeny graphs of degree-bounded digraphs to extend their work.

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arXiv2024

On $(1,2)$-step competition graphs of multipartite tournaments

A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph, denoted by $C_{1,2}(D)$, of $D$. In this paper, we completely characterize $C_{1,2}(D)$ for a tight multipartite tournament $D$. We will study $C_{1,2}(D)$ for a multipartite tournament $D$ that is not tight in a follow up paper.

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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.

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arXiv2024

On $(1,2)$-step competition graphs of multipartite tournaments II

A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph, denoted by $C_{1,2}(D)$, of $D$. In the previous paper titled "On $(1,2)$-step competition graphs of multipartite tournaments" \cite{choi202412step} we completely characterize $C_{1,2}(D)$ for a tight multipartite tournament $D$. As an extension, in this paper, we study $(1,2)$-step competition graphs of multipartite tournaments that are not tight, which will be called loose. For a loose multipartite tournament $D$, various meaningful results are obtained in terms of $C_{1,2}(D)$ being interval and $C_{1,2}(D)$ being connected.

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arXiv2023

Row graphs of Toeplitz matrices

In this paper, we study row graphs of Toeplitz matrices. The notion of row graphs was introduced by Greenberg et al. in 1984 and is closely related to the notion of competition graphs, which has been extensively studied since Cohen had introduced it in 1968. To understand the structure of the row graphs of Toeplitz matrices, which seem to be quite complicated, we have begun with Toeplitz matrices whose row graphs are triangle-free. We could show that if the row graph G of a Toeplitz matrix T is triangle-free, then T has the maximum row sum at most 2. Furthermore, it turns out that G is a disjoint union of paths and cycles whose lengths cannot vary that much in such a case. Then we study (0, 1)-Toeplitz matrices whose row graphs have only path components, only cycle components, and a cycle component of specific length, respectively. In particular, we completely characterize a (0, 1)-Toeplitz matrix whose row graph is a cycle.

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arXiv2022

Competitively orientable complete multipartite graphs

We say that a digraph $D$ is competitive if any pair of vertices has a common out-neighbor in $D$ and that a graph $G$ is competitively orientable if there exists a competitive orientation of $G$. The notion of competitive digraphs arose while studying digraph whose competition graphs are complete. We derive some useful properties of competitively orientable graphs and show that a complete graph of order $n$ is competitively orientable if and only if $n \geq 7$. Then we completely characterize a competitively orientable complete multipartite graph in terms of the sizes of its partite sets. Moreover, we present a way to build a competitive multipartite tournament in each of competitively orientable cases.

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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.

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arXiv2022

Digraphs whose m-step competition graphs are trees

In this paper, we completely characterize the digraphs of order $n$ whose $m$-step competition graphs are star graphs for positive integers $2\leq m < n$. This result in matrix version identifies the solution set to the matrix equation $X^m(X^T)^m= Λ_n+I_n$ for positive integers $2\leq m < n$ where $I_n$ is the identity matrix of order $n$ and $Λ_n$ is a $(0,1)$ Boolean matrix such that the first row and the first column consist of $1$'s except $(1,1)$-entry and the remaining entries are $0$, which is the adjacency matrix of a star graph of order $n$. We also derive meaningful properties of the digraphs whose $m$-step competition graphs are trees. In the process, we extend a result of Helleloid~[Connected triangle-free $m$-step competition graphs, Discrete Appl.\ Math.\ 145 (2005) 376--383] by showing that for all positive integers $m \geq 2$ and $n$, the connected triangle-free $m$-step competition graph on $n$ vertices is a tree.

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arXiv2022

Planarity of generalized ladder graphs

The Cartesian product of P_2 and P_n is called an n-ladder graph for a positive integer n. We call two paths P_m and P_n together with some edges each of which joins a vertex on P_m and a vertex on P_n a generalized (m,n)-ladder graph. In this paper, we completely characterize the planar generalized ladder graphs and the outerplanar generalized ladder graphs. A functigraph C(P_n ,f) is a generalized (n,n)-ladder graph. Consequently, our result solves the problem posed by A. Chen et al. (2011) to characterize planar functigraphs C(P_n ,f).

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