• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.637-646
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• 2011

Cho and Kim [4] and Kim [6] introduced the concept of the competition index of a digraph. Cho and Kim [4] and Akelbek and Kirkland [1] also studied the upper bound of competition indices of primitive digraphs. In this paper, we study the upper bound of competition indices of strongly connected digraphs. We also study the relation between competition index and ordinary index for a symmetric strongly connected digraph.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.1
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• pp.65-76
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• 2020

The notion of m-step competition graph was introduced by Cho et al. in 2000 as an interesting variation of competition graph. In this paper, we study the m-step competition graphs of d-partial orders, which generalizes the results obtained by Park et al. in 2011 and Choi et al. in 2018.

• Journal of the Korean School Mathematics Society
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• v.9 no.4
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• pp.481-495
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• 2006

In this paper we analyze creative mathematical competition for Russian mathematics teachers, and try to find out some suggestions for developing mathematics teacher's professional abilities. For these purposes we analyze problems and result data on the creative mathematical competition(for example, mean, distribution, reliability, discrimination, difficulty), and extract some meanings related with expanding teacher's professional abilities and activating studies related with mathematical pedagogy.

• Research in Mathematical Education
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• v.12 no.4
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• pp.271-281
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• 2008

Quarter a century ago, I founded the Colorado Mathematical Olympiad. The Colorado Mathematical Olympiad is the largest essay-type in-person mathematical competition in the United States, with 600 to 1,000 participants competing annually for prizes. In this article, I explain what it is, how it works, give examples of problems and solutions, and share with the reader careers of some of the Olympiad's winners.

• Bulletin of the Korean Mathematical Society
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• v.45 no.2
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• pp.385-396
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• 2008

For a positive integer m and a digraph D, the m-step competition graph $C^m$ (D) of D has he same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that there are directed walks of length m from u to x and from v to x. Cho and Kim [6] introduced notions of competition index and competition period of D for a strongly connected digraph D. In this paper, we extend these notions to a general digraph D. In addition, we study competition indices of tournaments.

• Journal of the Korean Mathematical Society
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• v.48 no.4
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• pp.691-702
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• 2011

The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between 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 defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.2001-2011
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• 2013

Cho and Kim [4] have introduced the concept of the competition index of a digraph. Similarly, the competition index of an $n{\times}n$ Boolean matrix A is the smallest positive integer q such that $A^{q+i}(A^T)^{q+i}=A^{q+r+i}(A^T)^{q+r+i}$ for some positive integer r and every nonnegative integer i, where $A^T$ denotes the transpose of A. In this paper, we study the upper bound of the competition index of a Boolean matrix. Using the concept of Boolean rank, we determine the upper bound of the competition index of a nearly reducible Boolean matrix.

• Bulletin of the Korean Mathematical Society
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• v.49 no.6
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• pp.1255-1262
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• 2012

A type of delayed Lotka-Volterra competition reaction-diffusion system is considered. By constructing a new Lyapunov function, we prove that the unique positive steady-state solution is globally asymptotically stable when interspecies competition is weaker than intraspecies competition. Moreover, we show that the stability property does not depend on the diffusion coefficients and time delays.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.649-657
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• 2000

The 2-step competition graph of D has the same vertex set as D and an edge between vertices x and y if and only if there exist (x, z)-walk of length 2 and (y, z)-walk of length 2 for some vertex z in D. The 2-step competition number of a graph G is the smallest number k such that G together with k isolated vertices is the 2-step competition graph of an acyclic digraph. Cho, et al. showed that the 2-step competition number of a path of length at least two is two. In this paper, we characterize all the minimal acyclic digraphs whose 2-step competition graphs are paths of length n with two isolated vertices and construct all such digraphs.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1251-1264
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• 2005

Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u, x) and (v, x) are arcs of D. The competition number of a graph G, denoted by k(G), is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. It is known to be difficult to compute the competition number of a graph in general. Even characterizing the graphs with competition number one looks hard. In this paper, we continue the work done by Cho and Kim[3] to characterize the graphs with one hole and competition number one. We give a sufficient condition for a graph with one hole to have competition number one. This generates a huge class of graphs with one hole and competition number one. Then we completely characterize the graphs with one hole and competition number one that do not have a vertex adjacent to all the vertices of the hole. Also we show that deleting pendant vertices from a connected graph does not change the competition number of the original graph as long as the resulting graph is not trivial, and this allows us to construct infinitely many graph having the same competition number. Finally we pose an interesting open problem.