• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.11 no.3
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• pp.1336-1356
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• 2017

The spectrum scarcity crisis has resulted in a shortage of resources for many emerging wireless services, and research on dynamic spectrum management has been used to solve this problem. Game theory can allocate resources to users in an economic way through market competition. In this paper, we propose a bidding game-based spectrum allocation mechanism in cognitive radio network. In our framework, primary networks provide heterogeneous wireless service and different numbers of channels, while secondary users have diverse bandwidth demands for transmission. Considering the features of traffic and QoS demands, we design a weighted interference graph-based grouping algorithm to divide users into several groups and construct the non-interference user-set in the first step. In the second step, we propose the dynamic bidding game-based spectrum allocation strategy; we analyze both buyer's and seller's revenue and determine the best allocation strategy. We also prove that our mechanism can achieve balanced pricing schema in competition. Theoretical and simulation results show that our strategy provides a feasible solution to improve spectrum utilization, can maximize overall utility and guarantee users' individual rationality.

• KIEE International Transactions on Power Engineering
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• v.5A no.1
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• pp.79-84
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• 2005

In many countries, the electric power industry is undergoing significant changes known as deregulation and restructuring. These alterations introduce competition in generation and retail and require open access to the transmission network. The competition of the electric power industry causes many issues to surface. Among them, unbundling of the transmission service is probably the most complicated as it is a single and integrated sector and the transmission revenue requirement must be allocated to market participants in a fair way. In these situations, it is valuable to research the methodologies to allocate transmission usage. The power tracing method offers useful information such as which generators supply a particular load or how much each generator (load) uses a particular transmission line. With this information, we can allocate required transmission revenue to market participants. Recently, several algorithms were proposed for tracing power flow but there is no dominant power tracing method. This paper proposes a power tracing method based on graph theory and complex-current distribution. For practicability, the proposed method for transmission usage allocation is applied to IEEE 30 buses and compared with the method proposed by Felix F.Wu.

• Proceedings of the KIEE Conference
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• 2002.07a
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• pp.422-424
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• 2002

Many methods about real power flow tracing have been suggested. Electric power industrials and organizations of the world use the method which is best suitable to themselves in practical aspects. In this paper we calculate the real power transfer between individual generators and loads referencing the method introduced by oversea's paper. It is considered to be significant to the wholesale competition market and transmission open access. Based on ac load flow solution and graph theory, the simulation on IEEE 30-bus system are carried out and the results are compared with that of oversea's paper. Also the simulation on the power system of Korea is carried out and the results are analyzed.