arXiv subjects

Hojin Chu

Publications and source records attributed to Hojin Chu.

15 recordsLinked to original sources

arXiv2026

Connectivity keeping paths in digraphs

Mader conjectured that every $k$-strong digraph $D$ with minimum semidegree $\delta^0(D)\ge 2k+m-1$ contains a dipath $P$ of order $m$ such that $D-V(P)$ remains $k$-strong. For $k=1$, he obtained the weaker bound $\delta^0(D)\ge 2m$. We confirm the conjecture for $k=1$ by showing that the sharp bound $\delta^0(D)\ge m+1$ suffices. As a consequence, we show that for every integer $m\ge2$, every strongly connected digraph $D$ with $\delta^0(D)\ge\max\{2,m-1\}$ contains a dipath $P$ of order $m$ such that $D-A(P)$ is strongly connected.

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arXiv2026

A sharp extension of Halin's removable-edge theorem to matchings

A subgraph $H$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-E(H)$ remains $k$-connected. Halin proved that every $k$-connected graph $G$ with $\delta(G)\ge k+1$ has a $k$-removable edge. We extend this result from a single edge to matchings of any prescribed size by showing that, for positive integers $k$ and $m$, every $k$-connected graph $G$ with $\delta(G)\ge\max\{k+1,2m-2\}$ contains a $k$-removable matching of size $m$, unless $G\cong K_{2m-1}$, or $(k,m)=(1,2)$ and $G$ is a cycle. This confirms a conjecture of Li, Zhou, Fujita, and Mao. The minimum degree bound is sharp, and both exceptions are unavoidable. Consequently, $\max\{k+1,2m-1\}$ is the sharp minimum degree threshold guaranteeing such a matching without exceptions. The proof combines a prescribed-set strengthening of Halin's removable-edge theorem with an extremal analysis of maximum $k$-removable matchings.

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arXiv2026

Minimum degree conditions for removable matchings in $k$-connected graphs

In 1969, Halin proved that every $k$-connected graph $G$ with minimum degree at least $k+1$ contains an edge $e$ such that $G-e$ is $k$-connected. As an edge is a matching of size one, it is natural to ask whether Halin's result extends to matchings of larger size, a question recently investigated by Li, Zhou, Fujita, and Mao. A matching $M$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-M$ is $k$-connected. In this paper, we study minimum degree conditions that guarantee the existence of a $k$-removable matching of prescribed size. Specifically, we prove that for all positive integers $k$ and $m$, every $k$-connected graph $G$ with at least $2m$ vertices contains a $k$-removable matching of size $m$ if \[\delta(G)\ \ge\ \begin{cases} \max\bigl\{k+\bigl\lceil\tfrac m2\bigr\rceil,\ 2m\bigr\} & \text{if } k\ge m,\\[2pt] k+m & \text{if } k<m. \end{cases}\] As a consequence, every $k$-connected graph $G$ with $\delta(G)\ge2k+1$ contains a $k$-removable matching of size $\bigl\lceil(\delta(G)+1)/2\bigr\rceil$, unless $\delta(G)$ is even and $G\cong K_{\delta(G)+1}$. This verifies a conjecture of Li, Zhou, Fujita, and Mao in the range $\delta(G)\ge2k+1$. Our main tool, of independent interest, is a strengthening of Halin's result producing a $k$-removable edge that avoids a prescribed set of vertices.

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arXiv2026

On 2-connected graphs without cycles of length 1 modulo 3

Burr and Erd\H{o}s conjectured in 1976 that for all integers $k>\ell\geq 0$ such that $k\mathbb{Z}+\ell$ contains an even integer, every $n$-vertex graph without cycles of length $\ell$ modulo $k$ has at most a linear number of edges in $n$. Bollob\'{a}s confirmed the conjecture in 1977, and Erd\H{o}s further asked for the exact extremal number. To the best of our knowledge, this problem has been solved only for all residues when $k\leq 4$, and for $\ell\in \{0,2\}$ when $k\geq 5$ is odd. In particular, Bai {\it et al.} [arXiv:2503.03504] proved that if $G$ is an $n$-vertex graph with no cycles of length $1$ modulo $3$, then $e(G)\le \frac{5}{3}(n-1)$, and when $9\mid (n-1)$ the equality holds if and only if each block of $G$ is isomorphic to the Petersen graph. Note that for $n> 18$ every extremal graph contains a cut-vertex. In this paper, we investigate the 2-connected setting and determine the maximum number of edges in a 2-connected graph with no cycles of length $1$ modulo $3$. Our results provide a sharp extremal bound and a complete characterization of the extremal graphs, revealing structural differences from the general case. Combining this with the result of Bai {\it et al.}, we also obtain a complete characterization of all extremal graphs in the general setting, including the cases where $9\nmid (n-1)$. Finally, we determine the maximum number of edges in a $2$-connected graph with no cycles of length $2$ modulo $4$, whose extremal graphs differ substantially from those in the general setting. Consequently, the extremal numbers for $2$-connected graphs with no cycle of a fixed length modulo $k$ are now determined for all $k\leq 4$.

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arXiv2026

Existence of cycles of length divisible by 3 or 4

Dean conjectured that for each integer $k \ge 3$, every graph with minimum degree at least $k$ has a cycle whose length is divisible by $k$; this conjecture is known to be true for all $k\neq 5$. For $k\in\{3,4\}$, stronger statements are true: every graph with minimum degree at least $2$ and at most $k-2$ vertices of degree $2$ has a cycle whose length is divisible by $k$. We further strengthen these results by characterizing all graphs with minimum degree at least $2$ and at most three vertices of degree $2$ that have no cycle of length divisible by $k$, for each $k\in\{3,4\}$. As a corollary, we obtain that every graph with minimum degree at least $2$ and at most two vertices of degree $2$ has a cycle whose length is divisible by $3$, and that every graph on at least nine vertices with minimum degree at least $2$ and at most three vertices of degree $2$ has a cycle whose length is divisible by $4$.

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arXiv2025

Connectivity keeping trees in triangle-free graphs

In 2012, Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor+m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2022, Luo, Tian, and Wu considered an analogous problem for bipartite graphs and conjectured that for any tree $T$ with bipartition $(X,Y)$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+\max\{|X|,|Y|\}$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In this paper, we relax the bipartite assumption by considering triangle-free graphs and prove that for any tree $T$ of order $m$, every $k$-connected triangle-free graph $G$ with minimum degree at least $2k+3m-4$ contains a subtree $T' \cong T$ such that $G-V(T')$ remains $k$-connected. Furthermore, we establish refined results for specific subclasses such as bipartite graphs or graphs with girth at least five.

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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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arXiv2025

On $2$-connected graphs avoiding cycles of length $0$ modulo $4$

For two integers $k$ and $\ell$, an $(\ell \text{ mod }k)$-cycle means a cycle of length $m$ such that $m\equiv \ell\pmod{k}$. In 1977, Bollobás proved a conjecture of Burr and Erdős by showing that if $\ell$ is even or $k$ is odd, then every $n$-vertex graph containing no $(\ell \text{ mod }k)$-cycles has at most a linear number of edges in terms of $n$. Since then, determining the exact extremal bounds for graphs without $(\ell \text{ mod }k)$-cycles has emerged as an interesting question in extremal graph theory, though the exact values are known only for a few integers $\ell$ and $k$. Recently, Győri, Li, Salia, Tompkins, Varga and Zhu proved that every $n$-vertex graph containing no $(0 \text{ mod }4)$-cycles has at most $\left\lfloor \frac{19}{12}(n -1) \right\rfloor$ edges, and they provided extremal examples that reach the bound, all of which are not $2$-connected. In this paper, we show that a $2$-connected graph without $(0 \text{ mod } 4)$-cycles has at most $\left\lfloor \frac{3n-1}{2} \right\rfloor$ edges, and this bound is tight by presenting a method to construct infinitely many extremal examples.

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arXiv2025

Linear-Time Computation of the Frobenius Normal Form for Symmetric Toeplitz Matrices via Graph-Theoretic Decomposition

We introduce a linear-time algorithm for computing the Frobenius normal form (FNF) of symmetric Toeplitz matrices by utilizing their inherent structural properties through a graph-theoretic approach. Previous results of the authors established that the FNF of a symmetric Toeplitz matrix is explicitly represented as a direct sum of symmetric irreducible Toeplitz matrices, each corresponding to connected components in an associated weighted Toeplitz graph. Conventional matrix decomposition algorithms, such as Storjohann's method (1998), typically have cubic-time complexity. Moreover, standard graph component identification algorithms, such as breadth-first or depth-first search, operate linearly with respect to vertices and edges, translating to quadratic-time complexity solely in terms of vertices for dense graphs like weighted Toeplitz graphs. Our method uniquely leverages the structural regularities of weighted Toeplitz graphs, achieving linear-time complexity strictly with respect to vertices through two novel reductions: the α-type reduction, which eliminates isolated vertices, and the β-type reduction, applying residue class contractions to achieve rapid structural simplifications while preserving component structure. These reductions facilitate an efficient recursive decomposition process that yields linear-time performance for both graph component identification and the resulting FNF computation. This work highlights how structured combinatorial representations can lead to significant computational gains in symbolic linear algebra.

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arXiv2024

Structural properties of a symmetric Toeplitz and Hankel matrices

In this paper, we investigate properties of a symmetric Toeplitz matrix and a Hankel matrix by studying the components of its graph. To this end, we introduce the notion of ``weighted Toeplitz graph" and ``weighted Hankel graph", which are weighted graphs whose adjacency matrix are a symmetric Toeplitz matrix and a Hankel matrix, respectively. By studying the components of a weighted Toeplitz graph, we show that the Frobenius normal form of a symmetric Toeplitz matrix is a direct sum of symmetric irreducible Toeplitz matrices. Similarly, by studying the components of a weighted Hankel matrix, we show that the Frobenius normal form of a Hankel matrix is a direct sum of irreducible Hankel matrices.

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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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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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