arXiv subjects

Boram Park

Publications and source records attributed to Boram Park.

At least 55 records · Page 3Linked to original sources

arXiv2018

On the weighted safe set problem on paths and cycles

Let $G$ be a graph, and let $w: V(G) \to \mathbb{R}$ be a weight function on the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\sum_{v \in X} w(v).$ 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 $s(G,w)$ and connected weighted safe number $cs(G,w)$ of $(G,w)$ are the minimum weights $w(S)$ among all weighted safe sets and all connected weighted safe sets of $(G,w)$, respectively. It is easy to see that for any pair $(G,w)$, ${s}(G,w) \le {cs}(G,w)$ by their definitions. In this paper, we discuss the possible equality when $G$ is a path or a cycle. We also give an answer to a problem due to Tittmann et al. [Eur. J. Combin. Vol. 32 (2011)] concerning subgraph component polynomials for cycles and complete graphs.

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arXiv2018

Graph invariants and Betti numbers of real toric manifolds

For a graph $G$, a graph cubeahedron $\square_G$ and a graph associahedron $\triangle_G$ are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the $b$-number, and we show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to a graph cubeahedron. The $b$-number is a counterpart of the notion of $a$-number, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold $X^\mathbb{R}(\triangle_G)$ corresponding to a graph associahedron. We also study various relationships between $a$-numbers and $b$-numbers from a toric topological view. Interestingly, for a forest $G$ and its line graph $L(G)$, the real toric manifolds $X^\mathbb{R}(\triangle_G)$ and $X^\mathbb{R}(\square_{L(G)})$ have the same Betti numbers.

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arXiv2018

On incidence choosability of cubic graphs

An incidence of a graph $G$ is a pair $(u,e)$ where $u$ is a vertex of $G$ and $e$ is an edge of $G$ incident with $u$. Two incidences $(u,e)$ and $(v,f)$ of $G$ are adjacent whenever (i) $u=v$, or (ii) $e=f$, or (iii) $uv=e$ or $uv=f$. An incidence $k$-coloring of $G$ is a mapping from the set of incidences of $G$ to a set of $k$ colors such that every two adjacent incidences receive distinct colors. The notion of incidence coloring has been introduced by Brualdi and Quinn Massey (1993) from a relation to strong edge coloring, and since then, attracted by many authors. On a list version of incidence coloring, it was shown by Benmedjdoub et. al. (2017) that every Hamiltonian cubic graph is incidence 6-choosable. In this paper, we show that every cubic (loopless) multigraph is incidence 6-choosable. As a direct consequence, it implies that the list strong chromatic index of a $(2,3)$-bipartite graph is at most 6, where a (2,3)-bipartite graph is a bipartite graph such that one partite set has maximum degree at most 2 and the other partite set has maximum degree at most 3.

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arXiv2018

On shellability for a poset of even subgraphs of a graph

Given a simple graph $G$, a poset of its even subgraphs was firstly considered by S. Choi and H. Park to study the topology of a real toric manifold associated with $G$. S. Choi and the authors extended this to a graph allowing multiple edges, motivated by the work on the pseudograph associahedron of Carr, Devadoss and Forcey. In this paper, we completely characterize the graphs (allowing multiple edges) whose posets of even subgraphs are always shellable. By the result, we also compute the Betti numbers of a real toric manifold corresponding to a path with two multiple edges.

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arXiv2017

List 3-dynamic coloring of graphs with small maximum average degree

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r,°_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $χ_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The {\em list $r$-dynamic chromatic number} of a graph $G$ is denoted by $ch_r^d(G)$. Recently, Loeb et al. [UI] showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng et al. [Lai-16] studied the maximum average condition to have $χ_3^d (G) \leq 4, \ 5$, or $6$. On the other hand, Song et al. [SLW] showed that if $G$ is planar with girth at least 6, then $χ_r^d(G)\le r+5$ for any $r\ge 3$. In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $ch^d_3(G) \leq 6$ if $mad(G) < \frac{18}{7}$, $ch^d_3(G) \leq 7$ if $mad(G) < \frac{14}{5}$, and $ch^d_3(G) \leq 8$ if $mad(G) < 3$. All of the bounds are tight.

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arXiv2016

Cycles with two blocks in $k$-chromatic digraphs

Let $k$ and $\ell$ be positive integers. A cycle with two blocks $c(k,\ell)$ is an oriented cycle which consists of two internally (vertex) disjoint directed paths of lengths at least $k$ and $\ell$, respectively, from a vertex to another one. A problem of Addario-Berry, Havet and Thomassé (2007) asked if, given positive integers $k$ and $\ell$ such that $k+\ell\ge 4$, any strongly connected digraph $D$ containing no $c(k,\ell)$ has chromatic number at most $k+\ell-1$. In this paper, we show that such digraph $D$ has chromatic number at most $O((k+\ell)^2)$, improving the previous upper bound $O((k+\ell)^4)$ obtained by Cohen, Havet, Lochet and Nisse (2016). In fact, we are able to find a digraph which shows that the answer to the above problem is no. We also show that if in addition $D$ is Hamiltonian, then its underlying simple graph is $(k+\ell-1)$-degenerate and thus the chromatic number of $D$ is at most $k+\ell$, which is tight.

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arXiv2016

Pseudograph and its associated real toric manifold

Given a simple graph $G$, the graph associahedron $P_G$ is a convex polytope whose facets correspond to the connected induced subgraphs of $G$. Graph associahedra have been studied widely and are found in a broad range of subjects. Recently, S. Choi and H. Park computed the rational Betti numbers of the real toric variety corresponding to a graph associahedron under the canonical Delzant realization. In this paper, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss and Forcey, and then discuss how to compute the Poincaré polynomial of the real toric variety corresponding to a pseudograph associahedron under the canonical Delzant realization.

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arXiv2016

The Betti numbers of real toric varieties associated to Weyl chambers of type $B$

We compute the (rational) Betti number of real toric varieties associated to Weyl chambers of type $B$. Furthermore, we show that their integral cohomology is $p$-torsion free for all odd primes $p$.

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arXiv2015

On the safe set of Cartesian product of two complete graphs

For a connected graph $G$, a vertex subset $S$ of $V(G)$ is a safe set if for every component $C$ of the subgraph of $G$ induced by $S$, $|C| \ge |D|$ holds for every component $D$ of $G-S$ such that there exists an edge between $C$ and $D$, and, in particular, if the subgraph induced by $S$ is connected, then $S$ is called a connected safe set. For a connected graph $G$, the safe number and the connected safe number of $G$ are the minimum among sizes of the safe sets and the minimum among sizes of the connected safe sets, respectively, of $G$. Fujita et al. introduced these notions in connection with a variation of the facility location problem. In this paper, we study the safe number and the connected safe number of Cartesian product of two complete graphs. Figuring out a way to reduce the number of components to two without changing the size of safe set makes it sufficient to consider only partitions of an integer into two parts without which it would be much more complicated to take care of all the partitions. In this way, we could show that the safe number and the connected safe number of Cartesian product of two complete graphs are equal and present a polynomial-time algorithm to compute them. Especially, in the case where one of complete components has order at most four, we precisely formulate those numbers.

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arXiv2015

Coloring the squares of graphs whose maximum average degrees are less than 4

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. The {\em maximum average degree} of $G$, $mad (G)$, is the maximum among the average degrees of the subgraphs of $G$. It is known in \cite{BLP-14-JGT} that there is no constant $C$ such that every graph $G$ with $mad(G)< 4$ has $χ(G^2) \leq Δ(G) + C$. Charpentier \cite{Charpentier14} conjectured that there exists an integer $D$ such that every graph $G$ with $Δ(G)\ge D$ and $mad(G)<4$ has $χ(G^2) \leq 2 Δ(G)$. Recent result in \cite{BLP-DM} implies that $χ(G^2) \leq 2 Δ(G)$ if $mad(G) < 4 -\frac{1}{c}$ with $Δ(G) \geq 40c -16$. In this paper, we show for $c\ge 2$, if $mad(G) < 4 - \frac{1}{c}$ and $Δ(G) \geq 14c-7$, then $χ_\ell(G^2) \leq 2 Δ(G)$, which improves the result in \cite{BLP-DM}. We also show that for every integer $D$, there is a graph $G$ with $Δ(G)\ge D$ such that $mad(G)<4$, and $χ(G^2) \geq 2Δ(G) +2$, which disproves Charpentier's conjecture. In addition, we give counterexamples to Charpentier's another conjecture in \cite{Charpentier14}, stating that for every integer $k\ge 3$, there is an integer $D_k$ such that every graph $G$ with $mad(G)<2k$ and $Δ(G)\ge D_k$ has $χ(G^2) \leq kΔ(G) -k$.

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arXiv2014

Coloring of the square of Kneser graph $K(2k+r,k)$

The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of an $n$ elements set, with two vertices adjacent if they are disjoint. The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Determining the chromatic number of the square of the Kneser graph $K(n, k)$ is an interesting graph coloring problem, and is also related with intersecting family problem. The square of $K(2k, k)$ is a perfect matching and the square of $K(n, k)$ is the complete graph when $n \geq 3k-1$. Hence coloring of the square of $K(2k +1, k)$ has been studied as the first nontrivial case. In this paper, we focus on the question of determining $χ(K^2(2k+r,k))$ for $r \geq 2$. Recently, Kim and Park \cite{KP2014} showed that $χ(K^2(2k+1,k)) \leq 2k+2$ if $ 2k +1 = 2^t -1$ for some positive integer $t$. In this paper, we generalize the result by showing that for any integer $r$ with $1 \leq r \leq k -2$, (a) $χ(K^2 (2k+r, k)) \leq (2k+r)^r$, if $2k + r = 2^t$ for some integer $t$, and (b) $χ(K^2 (2k+r, k)) \leq (2k+r+1)^r$, if $2k + r = 2^t-1$ for some integer $t$. On the other hand, it was showed in \cite{KP2014} that $χ(K^2 (2k+r, k)) \leq (r+2)(3k + \frac{3r+3}{2})^r$ for $2 \leq r \leq k-2$. We improve these bounds by showing that for any integer $r$ with $2 \leq r \leq k -2$, we have $χ(K^2 (2k+r, k)) \leq2 \left(\frac{9}{4}k + \frac{9(r+3)}{8} \right)^r$. Our approach is also related with injective coloring and coloring of Johnson graph.

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arXiv2014

Bipartite graphs whose squares are not chromatic-choosable

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $χ(H)$ and $χ_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_{\ell} (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable for every graph $G$. Recently, Kim and Park showed that the List Square Coloring Conjecture does not hold in general by finding a family of graphs whose squares are complete multipartite graphs with partite sets of unbounded size. It is a well-known fact that the List Total Coloring Conjecture is true if the List Square Coloring Conjecture holds for special class of bipartite graphs. On the other hand, the counterexamples to the List Square Coloring Conjecture are not bipartite graphs. Hence a natural question is whether $G^2$ is chromatic-choosable or not for every bipartite graph $G$. In this paper, we give a bipartite graph $G$ such that $χ_{\ell} (G^2) \neq χ(G^2)$. Moreover, we show that the value $χ_{\ell}(G^2) - χ(G^2)$ can be arbitrarily large.

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arXiv2013

Chromatic-choosability of the power of graphs

The $k$th power $G^k$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^k$ if the distance between $u$ and $v$ in $G$ is at most $k$. Let $χ(H)$ and $χ_l(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_l (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. A natural question raised by Xuding Zhu (2012) is whether there exists a constant integer $k$ such that $G^k$ is chromatic-choosable for every graph $G$. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) asked whether $G^2$ is chromatic-choosable for every graph $G$. Kim and Park (2013) answered the Kostochka and Woodall's question in the negative by finding a family of graphs whose squares are complete multipartite graphs with partite sets of equal and unbounded size. In this paper, we answer Zhu's question by showing that for every integer $k \geq 2$, there exists a graph $G$ such that $G^k$ is not chromatic-choosable. Moreover, for any fixed $k$ we show that the value $χ_l(G^k) - χ(G^k)$ can be arbitrarily large.

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arXiv2013

The phylogeny graphs of doubly partial orders

The competition graph of a doubly partial order is known to be an interval graph. The CCE graph and the niche graph of a doubly partial order are also known to be interval graphs if the graphs do not contain a cycle of length four and three as an induced subgraph, respectively. Phylogeny graphs are variant of competition graphs. The phylogeny graph $P(D)$ of a digraph $D$ is the (simple undirected) graph defined by $V(P(D)):=V(D)$ and $E(P(D)):=\{xy \mid N^+_D(x) \cap N^+_D(y) \neq \emptyset \} \cup \{xy \mid (x,y) \in A(D) \}$, where $N^+_D(x):=\{v \in V(D) \mid (x,v) \in A(D)\}$. In this note, we show that the phylogeny graph of a doubly partial order is an interval graph. We also show that, for any interval graph $G$, there exists an interval graph $\tilde{G}$ such that $\tilde{G}$ contains the graph $G$ as an induced subgraph and that $\tilde{G}$ is the phylogeny graph of a doubly partial order.

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arXiv2013

Counterexamples to the List Square Coloring Conjecture

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arXiv2013

Improved bounds on the chromatic numbers of the square of Kneser graphs

The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-elements subsets of an $n$-element set, with two vertices adjacent if the sets are disjoint. The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Determining the chromatic number of the square of the Kneser graph $K(2k+1, k)$ is an interesting problem, but not much progress has been made. Kim and Nakprasit \cite{2004KN} showed that $χ(K^2(2k+1,k)) \leq 4k+2$, and Chen, Lih, and Wu \cite{2009CLW} showed that $χ(K^2(2k+1,k)) \leq 3k+2$ for $k \geq 3$. In this paper, we give improved upper bounds on $χ(K^2(2k+1,k))$. We show that $χ(K^2(2k+1,k)) \leq 2k+2$, if $ 2k +1 = 2^n -1$ for some positive integer $n$. Also we show that $χ(K^2(2k+1,k)) \leq \frac{8}{3}k+\frac{20}{3}$ for every integer $k\ge 2$. In addition to giving improved upper bounds, our proof is concise and can be easily understood by readers while the proof in \cite{2009CLW} is very complicated. Moreover, we show that $χ(K^2(2k+r,k))=Θ(k^r)$ for each integer $2 \leq r \leq k-2$.

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arXiv2012

Competitively tight graphs

The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$ and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ of a graph $G$ is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph $G$ is related to the edge clique cover number $θ_E(G)$ of the graph $G$ via $θ_E(G)-|V(G)|+2 \leq k(G) \leq θ_E(G)$. We first show that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there exists a graph $G$ with $k(G)=θ_E(G)-|V(G)|+m$ and characterize a graph $G$ satisfying $k(G)=θ_E(G)$. We then focus on what we call \emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e., $k(G)=θ_E(G)-|V(G)|+2$. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.

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arXiv2012

The competition number of a generalized line graph is at most two

In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this note, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.

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