arXiv subjects

Boram Park

Publications and source records attributed to Boram Park.

At least 37 records · Page 2Linked to original sources

arXiv2021

Tight bound for independent domination of cubic graphs without $4$-cycles

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $γ(G)$, is the minimum size of a dominating set of $G$. The independent domination number of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent. Recently, Abrishami and Henning proved that if $G$ is a cubic graph with girth at least $6$, then $i(G) \le \frac{4}{11}|V(G)|$. We show a result that not only improves upon the upper bound of the aforementioned result, but also applies to a larger class of graphs, and is also tight. Namely, we prove that if $G$ is a cubic graph without $4$-cycles, then $i(G) \le \frac{5}{14}|V(G)|$, which is tight. Our result also implies that every cubic graph $G$ without $4$-cycles satisfies $\frac{i(G)}{γ(G)} \le \frac{5}{4}$, which partially answers a question by O and West in the affirmative.

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arXiv2021

On independent domination of regular graphs

Given a graph $G$, a dominating set of $G$ is a set $S$ of vertices such that each vertex not in $S$ has a neighbor in $S$. The domination number of $G$, denoted $γ(G)$, is the minimum size of a dominating set of $G$. The independent domination number of $G$, denoted $i(G)$, is the minimum size of a dominating set of $G$ that is also independent. Note that every graph has an independent dominating set, as a maximal independent set is equivalent to an independent dominating set. Let $G$ be a connected $k$-regular graph that is not $K_{k, k}$ where $k\geq 4$. Generalizing a result by Lam, Shiu, and Sun, we prove that $i(G)\le \frac{k-1}{2k-1}|V(G)|$, which is tight for $k = 4$. This answers a question by Goddard et al. in the affirmative. We also show that $\frac{i(G)}{γ(G)} \le \frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph $G'$ with maximum degree at most $4$ satisfies $i(G') \le \frac{5}{9}|V(G')|$, which is also tight.

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arXiv2020

Improvements on Hippchen's Conjecture

Let $G$ be a $k$-connected graph on $n$ vertices. Hippchen's Conjecture states that two longest paths in $G$ share at least $k$ vertices. Gutiérrez recently proved the conjecture when $k\leq 4$ or $k\geq \frac{n-2}{3}$. We improve upon both results; namely, we show that two longest paths in $G$ share at least $k$ vertices when $k=5$ or $k\geq \frac{n+2}{5}$. This completely resolves two conjectures of Gutiérrez in the affirmative.

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arXiv2020

The size of graphs with restricted rainbow $2$-connection number

Let $k$ be a positive integer, and $G$ be a $k$-connected graph. An edge-coloured path is \emph{rainbow} if all of its edges have distinct colours. The \emph{rainbow $k$-connection number} of $G$, denoted by $rc_k(G)$, is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by $k$ internally vertex-disjoint rainbow paths. The function $rc_k(G)$ was introduced by Chartrand, Johns, McKeon and Zhang in 2009, and has since attracted significant interest. Let $t_k(n,r)$ denote the minimum number of edges in a $k$-connected graph $G$ on $n$ vertices with $rc_k(G)\le r$. Let $s_k(n,r)$ denote the maximum number of edges in a $k$-connected graph $G$ on $n$ vertices with $rc_k(G)\ge r$. The functions $t_1(n,r)$ and $s_1(n,r)$ have previously been studied by various authors. In this paper, we study the functions $t_2(n,r)$ and $s_2(n,r)$. We determine bounds for $t_2(n,r)$ which imply that $t_2(n,2)=(1+o(1))n\log_2 n$, and $t_2(n,r)$ is linear in $n$ for $r\ge 3$. We also provide some remarks about the function $s_2(n,r)$.

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arXiv2020

Decomposing planar graphs into graphs with degree restrictions

Given a graph $G$, a decomposition of $G$ is a partition of its edges. A graph is $(d, h)$-decomposable if its edge set can be partitioned into a $d$-degenerate graph and a graph with maximum degree at most $h$. For $d \le 4$, we are interested in the minimum integer $h_d$ such that every planar graph is $(d,h_d)$-decomposable. It was known that $h_3 \le 4$, $h_2\le 8$, and $h_1 = \infty$. This paper proves that $h_4=1, h_3=2$, and $4 \le h_2 \le 6$.

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arXiv2020

On toric ideals arising from signed graphs

A signed graph is a pair $(G,τ)$ of a graph $G$ and its sign $τ$, where a \textit{sign} $τ$ is a function from $\{ (e,v)\mid e\in E(G),v\in V(G), v\in e\}$ to $\{1,-1\}$. Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal $I_{(G,τ)}$ associated with a signed graph $(G,τ)$, and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of $I_{(G,τ)}$, and then focus on the complete intersection property. More precisely, we find a complete list of graphs $G$ such that $I_{(G,τ)}$ is a complete intersection for every sign $τ$.

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arXiv2020

The strong clique number of graphs with forbidden cycles

Given a graph $G$, the strong clique number of $G$, denoted $ω_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned Erdős--Nešetřil conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique number, and conjectured a quadratic upper bound in terms of the maximum degree. Recently, Cames van Batenburg, Kang, and Pirot conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles. Namely, if $G$ is a $C_{2k}$-free graph, then $ω_S(G)\leq (2k-1)Δ(G)-{2k-1\choose 2}$, and if $G$ is a $C_{2k}$-free bipartite graph, then $ω_S(G)\leq kΔ(G)-(k-1)$. We prove the second conjecture in a stronger form, by showing that forbidding all odd cycles is not necessary. To be precise, we show that a $\{C_5, C_{2k}\}$-free graph $G$ with $Δ(G)\ge 1$ satisfies $ω_S(G)\leq kΔ(G)-(k-1)$, when either $k\geq 4$ or $k\in \{2,3\}$ and $G$ is also $C_3$-free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for $k\geq 3$, we prove that a $C_{2k}$-free graph $G$ with $Δ(G)\ge 1$ satisfies $ω_S(G)\leq (2k-1)Δ(G)+(2k-1)^2$. This improves some results of Cames van Batenburg, Kang, and Pirot.

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arXiv2020

Partitioning planar graphs without $4$-cycles and $5$-cycles into bounded degree forests

In 1976, Steinberg conjectured that planar graphs without $4$-cycles and $5$-cycles are $3$-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For integers $d_1$ and $d_2$ satisfying $d_1+d_2\geq8$ and $d_2\geq d_1\geq 2$, it is known that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where each $V_i$ induces a graph with maximum degree at most $d_i$. Since Steinberg's Conjecture is false, a partition of $V(G)$ into two sets, where one induces an empty graph and the other induces a forest is not guaranteed. Our main theorem is at the intersection of the two aforementioned research directions. We prove that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where $V_1$ induces a forest with maximum degree at most $3$ and $V_2$ induces a forest with maximum degree at most $4$; this is both a relaxation of Steinberg's conjecture and a strengthening of results by Sittitrai and Nakprasit (2019) in a much stronger form.

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arXiv2020

The optimal proper connection number of a graph with given independence number

An edge-colored connected graph $G$ is properly connected if between every pair of distinct vertices, there exists a path that no two adjacent edges have a same color. Fujita (2019) introduced the optimal proper connection number ${\mathrm{pc}_{\mathrm{opt}}}(G)$ for a monochromatic connected graph $G$, to make a connected graph properly connected efficiently. More precisely, ${\mathrm{pc}_{\mathrm{opt}}}(G)$ is the smallest integer $p+q$ when one converts a given monochromatic graph $G$ into a properly connected graph by recoloring $p$ edges with $q$ colors. In this paper, we show that ${\mathrm{pc}_{\mathrm{opt}}}(G)$ has an upper bound in terms of the independence number $α(G)$. Namely, we prove that for a connected graph $G$, ${\mathrm{pc}_{\mathrm{opt}}}(G)\le \frac{5α(G)-1}{2}$. Moreoevr, for the case $α(G)\leq 3$, we improve the upper bound to $4$, which is tight.

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arXiv2020

Stable structure on safe set problems in vertex-weighted graphs

Let $G$ be a graph, and let $w$ be a positive real-valued weight function on $V(G)$. For every subset $S$ of $V(G)$, let $w(S)=\sum_{v \in S} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G-S$, we have $w(C) \geq w(D)$ whenever there is an edge between $C$ and $D$. If the subgraph of $G$ induced by a weighted safe set $S$ is connected, then the set $S$ is called a connected weighted safe set of $(G,w)$. The weighted safe number $\mathrm{s}(G,w)$ and connected weighted safe number $\mathrm{cs}(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. Note that for every pair $(G,w)$, $\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$ by their definitions. Recently, it was asked which pair $(G,w)$ satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs $G$ such that $\mathrm{s}(G,w)=\mathrm{cs}(G,w)$ for every weight function $w$ on $V(G)$.

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arXiv2020

Generalized list colouring of graphs

This paper disproves a conjecture of Wang, Wu, Yan and Xie, and answers in negative a question in Dvorak, Pekarek and Sereni. In return, we pose five open problems.

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arXiv2019

On Star 5-Colorings of Sparse Graphs

A \textit{star $k$-coloring} of a graph $G$ is a proper (vertex) $k$-coloring of $G$ such that the vertices on a path of length three receive at least three colors. Given a graph $G$, its \textit{star chromatic number}, denoted $χ_s(G)$, is the minimum integer $k$ for which $G$ admits a star $k$-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the \textit{maximum average degree}, denoted $mad(G)$, of a graph $G$ is $\max\left\{ \frac{2|E(H)|}{|V(H)|}:{H \subset G}\right\}$. It is known that for a graph $G$, if $mad(G)<\frac{8}{3}$, then $χ_s(G)\leq 6$, and if $mad(G)< \frac{18}{7}$ and its girth is at least 6, then $χ_s(G)\le 5$. We improve both results by showing that for a graph $G$, if $mad(G)\le \frac{8}{3}$, then $χ_s(G)\le 5$. As an immediate corollary, we obtain that a planar graph with girth at least 8 has a star 5-coloring, improving the best known girth condition for a planar graph to have a star 5-coloring.

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arXiv2019

On induced saturation for paths

For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an $H$-induced-saturated graph does not necessarily exist. In fact, Martin and Smith (2012) showed that $P_4$-induced-saturated graphs do not exist, where $P_k$ denotes a path on $k$ vertices. Axenovich and Csikós (2019) asked the existence of $P_k$-induced-saturated graphs for $k \ge 5$; it is easy to construct such graphs when $k\in\{2, 3\}$. Recently, Räty constructed a graph that is $P_6$-induced-saturated. In this paper, we show that there exists a $P_{k}$-induced-saturated graph for infinitely many values of $k$. To be precise, we find a $P_{3n}$-induced-saturated graph for every positive integer $n$. As a consequence, for each positive integer $n$, we construct infinitely many $P_{3n}$-induced-saturated graphs. We also show that the Kneser graph $K(n,2)$ is $P_6$-induced-saturated for every $n\ge 5$.

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arXiv2019

Collapsibility of non-cover complexes of graphs

Given a graph $G$, the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. Aharoni asked if the non-cover complex of a graph $G$ without isolated vertices is $(|V(G)|-i γ(G)-1)$-collapsible where $i γ(G)$ denotes the independent domination number of $G$. Extending a result by the second author, who verified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs. Namely, we show that for a graph $G$, the non-cover complex of a graph $G$ is $(|V(G)|-i γ(G)-1)$-collapsible.

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arXiv2019

Largest 2-regular subgraphs in 3-regular graphs

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs.

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arXiv2019

Maximum $k$-sum $\mathbf{n}$-free sets of the 2-dimensional integer lattice

For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a \textit{$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$ is a subset $S$ of $[n]\times [n]$ such that there are no elements ${\mathbf{a}}_1, \ldots, {\mathbf{a}}_k$ in $S$ satisfying ${\mathbf{a}}_1+\cdots+{\mathbf{a}}_k =\mathbf{b}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, we determine the maximum density of a {$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions.

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arXiv2018

Characterization of forbidden subgraphs for bounded star chromatic number

The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbids a bicolored subgraph $H$. When considering the class $\mathcal C$ of graphs with no $F$ as an induced subgraph, it is not hard to see that every graph in $\mathcal C$ has bounded chromatic number if and only if $F$ is a complete graph of size at most two. We study this phenomena for the class of graphs with no $F$ as a subgraph for $H$-avoiding coloring. We completely characterize all graphs $F$ where the class of graphs with no $F$ as a subgraph has bounded $H$-avoiding chromatic number for a large class of graphs $H$. As a corollary, our main result implies a characterization of graphs $F$ where the class of graphs with no $F$ as a subgraph has bounded star chromatic number. We also obtain a complete characterization for the acyclic chromatic number.

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arXiv2018

A Ramsey-type theorem for the matching number regarding connected graphs

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter $ρ$, every graph $G$ with sufficiently large $ρ(G)$ contains a `well-structured' induced subgraph $H$ with large $ρ(H)$. The classical Ramsey's theorem deals with the case when the graph parameter under consideration is the number of vertices; there is also a Ramsey-type theorem regarding connected graphs. Given a graph $G$, the matching number and the induced matching number of $G$ is the maximum size of a matching and an induced matching, respectively, of $G$. In this paper, we formulate Ramsey-type theorems for the matching number and the induced matching number regarding connected graphs. Along the way, we obtain a Ramsey-type theorem for the independence number regarding connected graphs as well.

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