Jihoon Choi — Page 2

arXiv2018

The graph grabbing game on $\{0,1\}$-weighted graphs

The \emph{graph grabbing game} is a two-player game on a weighted connected graph in which two players, Alice and Bob, alternatively remove non-cut vertices one by one to gain the weights on them. Alice wins the game if she gains at least half of the total weights. In this paper, we show that on every connected even graph which does contain a fully spiked cycle as an induced subgraph, Alice always has a winning strategy with an arbitrary weight function whose codomain is $\{0,1\}$. In addition, we give a list of forbidden subgraph for the family of graphs on which Alice has a winning strategy with an arbitrary weight function whose codomain is $\{0,1\}$.

math.CO↗

arXiv2016

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

In this paper, we study $(1,2)$-step competition graphs of bipartite tournaments. A bipartite tournament means an orientation of a complete bipartite graph. We show that the $(1,2)$-step competition graph of a bipartite tournament has at most one non-trivial component or consists of exactly two complete components of size at least three and, especially in the former, the diameter of the nontrivial component is at most three if it exists. Based on this result, we show that, among the connected non-complete graphs which are triangle-free or the cycles of which are edge-disjoint, $K_{1,4}$ is the only graph that can be represented as the $(1,2)$-step competition graph of a bipartite tournament. We also completely characterize a complete graph and the disjoint union of two complete graphs, respectively, which can be represented as the $(1,2)$-step competition graph of a bipartite tournament. Finally we present the maximum number of edges and the minimum number of edges which the $(1,2)$-step competition graph of a bipartite tournament might have.

math.CO↗

arXiv2016

On the phylogeny graphs of degree-bounded digraphs

Hefner [K. A. S. Hefner, K. F. Jones, S. -R. Kim, R. J. Lundgren and F. S. Roberts: $(i,j)$ competition graphs, Discrete Applied Mathematics, 32, (1991) 241-262] characterized acyclic digraphs each vertex of which has inderee and outdegree at most two and whose competition graphs are interval. They called acyclic digraphs each vertex of which has inderee and outdegree at most two $(2,2)$ digraphs. In this paper, we study the phylogeny graphs of $(2,2)$ digraphs. Especially, we give a sufficient condition and necessary conditions for $(2,2)$ digraphs having chordal phylogeny graphs. Phylogeny graphs are also called moral graphs in Bayesian network theory. Our work is motivated by problems related to evidence propagation in a Bayesian network for which it is useful to know which acyclic digraphs have their moral graphs being chordal.

math.CO↗

arXiv2016

On the competition graphs of $d$-partial orders

In this paper, we study the competition graphs of $d$-partial orders and obtain their characterization which extends results given by Cho and Kim \cite{chokim} in 2005. We also show that any graph can be made into the competition graph of a $d$-partial order for some positive integer $d$ as long as adding isolated vertices is allowed. We then introduce the notion of the partial order competition dimension of a graph and study graphs whose partial order competition dimensions are at most three.

math.CO↗

arXiv2016

On the competition graphs of $d$-partial orders

In this paper, we study the competition graphs of $d$-partial orders and obtain their characterization which extends results given by Cho and Kim \cite{chokim} in 2005. We also show that any graph can be made into the competition graph of a $d$-partial order for some positive integer $d$ as long as adding isolated vertices is allowed. We then introduce the notion of the partial order competition dimension of a graph and study graphs whose partial order competition dimensions are at most three.

math.CO↗

arXiv2016

On the partial order competition dimensions of chordal graphs

Choi {\it et al.} [{J.~Choi, K.~S.~Kim, S.~-R.~Kim, J.~Y.~Lee, and Y.~Sano}: {On the competition graphs of $d$-partial orders}, \emph{Discrete Applied Mathematics} (2015), \texttt{http://dx.doi.org/10.1016/j.dam.2015.11.004}] introduced the notion of the partial order competition dimension of a graph. It was shown that complete graphs, interval graphs, and trees, which are chordal graphs, have partial order competition dimensions at most three. In this paper, we study the partial order competition dimensions of chordal graphs. We show that chordal graphs have partial order competition dimensions at most three if the graphs are diamond-free. Moreover, we also show the existence of chordal graphs containing diamonds whose partial order competition dimensions are greater than three.

math.CO↗

arXiv2016

On the partial order competition dimensions of chordal graphs

Choi {\it et al.} [{J.~Choi, K.~S.~Kim, S.~-R.~Kim, J.~Y.~Lee, and Y.~Sano}: {On the competition graphs of $d$-partial orders}, \emph{Discrete Applied Mathematics} (2015), \texttt{http://dx.doi.org/10.1016/j.dam.2015.11.004}] introduced the notion of the partial order competition dimension of a graph. It was shown that complete graphs, interval graphs, and trees, which are chordal graphs, have partial order competition dimensions at most three. In this paper, we study the partial order competition dimensions of chordal graphs. We show that chordal graphs have partial order competition dimensions at most three if the graphs are diamond-free. Moreover, we also show the existence of chordal graphs containing diamonds whose partial order competition dimensions are greater than three.

math.CO↗

arXiv2014

A matrix equation $X^n = aI$

In this paper, we study a matrix equation $X^n = aI$. We factorize $X^n - aI$ based upon the factorization of $x^n - a$ and then give a necessary and sufficient condition for one of the factors to be the zero matrix.

math.RA↗

arXiv2013

On the matrix sequence $\{Γ(A^m)\}_{m=1}^\infty$ for a Boolean matrix $A$ whose digraph is linearly connected

In this paper, we extend the results given by Park {\em et al.} \cite{ppk} by studying the convergence of the matrix sequence $\{Γ(A^m)\}_{m=1}^\infty$ for a matrix $A \in \mathcal{B}_n$ the digraph of which is linearly connected with an arbitrary number of strong components. In the process for generalization, we concretize ideas behind their arguments. We completely characterize $A$ for which $\{Γ(A^m)\}_{m=1}^\infty$ converges. Then we find its limit when all of the irreducible diagonal blocks are of order at least two. We go further to characterize $A$ for which the limit of $\{Γ(A^m)\}_{m=1}^\infty$ is a $J$ block diagonal matrix. All of these results are derived by studying the $m$-step competition graph of the digraph of $A$.

math.CO↗

arXiv2013

Cryogenic cooling with cryocooler on a rotating system

We developed a system that continuously maintains a cryocooler for long periods on a rotating table. A cryostat that holds the cryocooler is set on the table. A compressor is located on the ground and supplies high-purity (> 99.999%) and high-pressure (1.7 MPa) helium gas and electricity to the cryocooler. The operation of the cryocooler and other instruments requires the development of interface components between the ground and rotating table. A combination of access holes at the center of the table and two rotary joints allows simultaneous circulation of electricity and helium gas. The developed system provides two innovative functions under the rotating condition; cooling from room temperature and the maintenance of a cold condition for long periods. We have confirmed these abilities as well as temperature stability under a condition of continuous rotation at 20 revolutions per minute. The developed system can be applied in various fields; e.g., in tests of Lorentz invariance, searches for axion, radio astronomy and cosmology, and application of radar systems. In particular, there is a plan to use this system for a radio telescope observing cosmic microwave background radiation.

astro-ph.IM↗