• Journal of the Korea Society of Computer and Information
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• v.19 no.5
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• pp.19-25
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• 2014

In this paper, I seek the chromatic number, the maximum number of colors necessary when adjoining vertices in the plane separated apart at the distance of 1 shall receive distinct colors. The upper limit of the chromatic number has been widely accepted as $4{\leq}{\chi}(G){\leq}7$ to which Hadwiger-Nelson proposed ${\chi}(G){\leq}7$ and Soifer ${\chi}(G){\leq}9$ I firstly propose an algorithm that obtains the minimum necessary chromatic number and show that ${\chi}(G)=3$ is attainable by determining the chromatic number for Hadwiger-Nelson's hexagonal graph. The proposed algorithm obtains a chromatic number of ${\chi}(G)=4$ assuming a Hadwiger-Nelson's hexagonal graph of 12 adjoining vertices, and again ${\chi}(G)=4$ for Soifer's square graph of 8 adjoining vertices. assert. Based on the results as such that this algorithm suggests the maximum chromatic number of a planar graph is ${\chi}(G)=4$ using simple assigned rule of polynomial time complexity to color for a vertex with minimum degree.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 1993.04a
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• pp.4-4
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• 1993

• Proceedings of the Korean Information Science Society Conference
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• 1998.10b
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• pp.410-412
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• 1998

그래프 채색 기법(Graph Coloring)에 기반한 레지스터 할당기들은 간섭 그래프의 서로 다른 노드(node)에 같은 레지스터를 할당함으로써 복사 명령어를 없앤다. 본 논문은 이러한 기법 가운데 보수적 융합(Conservative Coalescing)이 레지스터 쌍을 융합하는데 단점이 있음을 지적하고 이러한 문제가 낙관적 레지스터 융합 기법(Optmistic Register Coalescing)에 의해 해결될 수 있음을 보인다.

• The Journal of the Korea Contents Association
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• v.8 no.5
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• pp.52-60
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• 2008

A simple Time Division Multiplex(TDM) switching system which has been widely in satellite networks provides any size of bandwidth for a number of low bandwidth subscribers by allocating proper number of time slots in a frame. In this paper, we propose a new approach based on graph coloring model for efficient time slot assignment algorithm in contrast to network flow model in previous works. When the frame length of an initial matrix of time slot requests is 2's power, this matrix is divided into two matrices of time slot requests using binary divide and conquer method based on the graph coloring model. This process is continued until resulting matrices of time slot requests are of length one. While the most efficient algorithm proposed in the literature has time complexity of $O(N^{4.5})$, the time complexity of the proposed algorithm is $O(NLlog_2L)$, where N is the number of input/output links and L is the number of time slot alloted to each link in the frame.

• Journal of KIISE:Software and Applications
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• v.26 no.5
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• pp.692-701
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• 1999

수퍼스칼라(superscalar)나 VLIW 와 같은 명령어 수준 병렬화(ILP) 프로세서의 성능을 극대화하는 과감한 명령어 스케쥴링은 소프트웨어 파이프라이닝과같은 스케쥴링 과정을 거치면서 일반적인 복사 명령어 제거 기법으로 없앨 수 없는 서로 간섭하는 복사 명령을 많이 만들어내는데 루프 내부에 생성된 이러한 복사명령은 적절한 루프 펼침을 수행하여 간섭관계를 없앰으로서 제거할 수 있다. 본 논문에서는 이와 같이 루프 펼침이 수행된 루프 내부의 복사명령을 제거하는 기법으로 그래프 컬러링 상에 구현한 낙관적 융합기법을 제안한다. 그래프 컬러링에서의 융합기법은 간선의 개수가 많은 노드를 만들어 낼수 있으므로 채색성에 부정적인 영향을 주는 것으로 알려져 왔으나 본 기법에서는 융합되는 노드에 동시에 간섭하는 노드의 간선의 수가 줄어드는 긍정적인 영향을 최대한 이용하여 채색성을 높이고 융합된 노드에 대한 실제 버림(spill)이 일어나는 경우 유효 범위 분절(live range splitting)을 통하여 버림의 부담을 최대한 줄이도록 하였으며 이를 VLIW 스케쥴링 된 SPEC 정수벤치마크 루프내부의 복사 명령 제거에 적용한 결과 제거 가능한 복사 명령의 99%를 제거하면서도 버림명령은 다른 융합 기법과 비교하여 가장 적게 발생하는 우수한 결과를 얻을수 있었다.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.14 no.5
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• pp.263-269
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• 2014

In this paper, I disprove Hadwiger conjecture of the vertex coloring problem, which asserts that "All $K_k$-minor free graphs can be colored with k-1 number of colors, i.e., ${\chi}(G)=k$ given $K_k$-minor." Pursuant to Hadwiger conjecture, one shall obtain an NP-complete k-minor to determine ${\chi}(G)=k$, and solve another NP-complete vertex coloring problem as a means to color vertices. In order to disprove Hadwiger conjecture in this paper, I propose an algorithm of linear time complexity O(V) that yields the exact solution to the vertex coloring problem. The proposed algorithm assigns vertex with the minimum degree to the Maximum Independent Set (MIS) and repeats this process on a simplified graph derived by deleting adjacent edges to the MIS vertex so as to finally obtain an MIS with a single color. Next, it repeats the process on a simplified graph derived by deleting edges of the MIS vertex to obtain an MIS whose number of vertex color corresponds to ${\chi}(G)=k$. Also presented in this paper using the proposed algorithm is an additional algorithm that searches solution of ${\chi}^{{\prime}{\prime}}(G)$, the total chromatic number, which also remains NP-complete. When applied to a $K_4$-minor graph, the proposed algorithm has obtained ${\chi}(G)=3$ instead of ${\chi}(G)=4$, proving that the Hadwiger conjecture is not universally applicable to all the graphs. The proposed algorithm, however, is a simple algorithm that directly obtains an independent set minor of ${\chi}(G)=k$ to assign an equal color to the vertices of each independent set without having to determine minors in the first place.

• Journal of the Korea Society of Computer and Information
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• v.18 no.5
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• pp.113-120
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• 2013

This paper proposes an algorithm that proves an NP-complete 4-color theorem by employing a linear time complexity where $O(n)$. The proposed algorithm accurately halves the vertex set V of the graph $G=(V_1,E_1)$ into the Maximum Independent Set (MIS) $\bar{C_1}$ and the Minimum Vertex Cover Set $C_1$. It then assigns the first color to $\bar{C_1}$ and the second to $\bar{C_2}$, which, along with $C_2$, is halved from the connected graph $G=(V_2,E_2)$, a reduced set of the remaining vertices. Subsequently, the third color is assigned to $\bar{C_3}$, which, along with $C_3$, is halved from the connected graph $G=(V_3,E_3)$, a further reduced set of the remaining vertices. Lastly, denoting $C_3$ as $\bar{C_4}$, the algorithm assigns the forth color to $\bar{C_4}$. The algorithm has successfully obtained the chromatic number ${\chi}(G)=4$ with 100% probability, when applied to two actual map and two planar graphs. The proposed "four color algorithm", therefore, could be employed as a general algorithm to determine four-color for planar graphs.

• Journal of the Korea Society of Computer and Information
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• v.18 no.11
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• pp.159-165
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• 2013

This paper proposes a O(E) polynomial-time algorithm that has been devised to simultaneously solve edge-coloring problem and graph classification problem both of which remain NP-complete. The proposed algorithm selects an edge connecting maximum and minimum degree vertices so as to determine the number of edge coloring ${\chi}^{\prime}(G)$. Determined ${\chi}^{\prime}(G)$ is in turn either ${\Delta}(G)$ or ${\Delta}(G)+1$. Eventually, the result could be classified as class 1 if ${\chi}^{\prime}(G)={\Delta}(G)$ and as category 2 if ${\chi}^{\prime}(G)={\Delta}(G)+1$. This paper also proves Vizing's planar graph conjecture, which states that 'all simple, planar graphs with maximum degree six or seven are of class one, closing the remaining possible case', which has known to be NP-complete.

• Journal of KIISE:Computer Systems and Theory
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• v.26 no.8
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• pp.999-1007
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• 1999

이 논문은 재귀원형군 G(2^m , 2^k )를 그래프 이론적 관점에서 고찰하고 정점이 서로소인 사이클과 그래프 invariant에 관한 위상 특성을 제시한다. 재귀원형군은 1 에서 제안된 다중 컴퓨터의 연결망 구조이다. 재귀원형군 {{{{G(2^m , 2^k )가 길이 사이클을 가질 필요 충분 조건을 구하고, 이 조건하에서 G(2^m , 2^k )는 가능한 최대 개수의 정점이 서로소이고 길이가l`인 사이클을 가짐을 보인다. 그리고 정점 및 에지 채색, 최대 클릭, 독립 집합 및 정점 커버에 대한 그래프 invariant를 분석한다.Abstract In this paper, we investigate recursive circulant G(2^m , 2^k ) from the graph theory point of view and present topological properties of G(2^m , 2^k ) concerned with vertex-disjoint cycles and graph invariants. Recursive circulant is an interconnection structure for multicomputer networks proposed in 1 . A necessary and sufficient condition for recursive circulant {{{{G(2^m , 2^k ) to have a cycle of lengthl` is derived. Under the condition, we show that G(2^m , 2^k ) has the maximum possible number of vertex-disjoint cycles of length l`. We analyze graph invariants on vertex and edge coloring, maximum clique, independent set and vertex cover.

• Journal of the Korea Society of Computer and Information
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• v.18 no.2
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• pp.69-76
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• 2013

Reader-to-reader interference in RFID system is occurred due to the use of limited number of frequencies, and this is the main cause of read rate reduction in the passive RFID tags. Therefore, in order to maximize the read rate under the circumstances of limited frequency resources, it is necessary to minimize the frequency interference among RFID readers. This paper presents a hybrid FDM/TDM constraint satisfaction problem models for frequency interference minimization problems of the RFID readers, and assigns optimal channels to each readers using conventional backtracking search algorithms. A depth first search based on backtracking are accomplished to find solutions of constraint satisfaction problems. At this moment, a variable ordering algorithm is very important to find a solution quickly. Variable ordering algorithms applied in the experiment are known as efficient in the graph coloring. To justify the performance of the proposed constraint satisfaction problem model, optimal channels for each readers in the passive UHF RFID system are allocated by using computer simulation satisfying various interference constraints.