• Journal of applied mathematics & informatics
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• 제39권1_2호
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• pp.13-29
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• 2021

For a graph G = (V, E), a vertex coloring (or, simply, a coloring) of G is a function C : V (G) → {1, 2, …, k} (using the non-negative integers {1, 2, …, k} as colors). We say that a coloring of a graph G is injective if for every vertex v ∈ V (G), all the neighbors of v are assigned with distinct colors. The injective chromatic number χi(G) of a graph G is the least k such that there is an injective k-coloring [6]. In this paper, we study a natural variation of the injective coloring problem: coloring the edges of a graph under the same constraints (alternatively, to investigate the injective chromatic number of line graphs), we define the k- injective edge coloring of a graph G as a mapping C : E(G) → {1, 2, …, k}, such that for every edge e ∈ E(G), all the neighbors edges of e are assigned with distinct colors. The injective chromatic index χ′in(G) of G is the least positive integer k such that G has k- injective edge coloring, exact values of the injective chromatic index of different families of graphs are obtained, some related results and bounds are established. Finally, we define the injective clique number ωin and state a conjecture, that, for any graph G, ωin ≤ χ′in(G) ≤ ωin + 2.

• 디지털융복합연구
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• 제20권2호
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• pp.345-351
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• 2022

단순 무방향 그래프 G 의 L(2,1)-coloring은 d(u,v)가 두 정점 사이의 거리일 때 두 가지 조건 (1) d(x,y) = 1 라면 |f(x)-f(y)|≥ 2, (2) d(x,y) = 2 라면 |f(x)-f(y)|≥ 1 을 만족하는 함수 f : V → [0,1,…,k]를 정의하는 것이다. 임의의 L(2,1)-coloring c 에 대하여 G 의 c-span 은 λ(c)=max{|c(u)-c(v)|| u,v∈V} 이며, L(2,1)-coloring number 인 λ(G)는 모든 가능한 c 에 대하여 λ(G) = min{λ(c)} 로 정의된다. 본 논문에서는 Harary의 정리에 기반하여 지름이 2인 그래프에 대하여 여그래프에 해밀턴 경로의 존재여부를 Tabu Search를 사용해 판단하고 이를 통해 λ(G)가 n(=|V|)과 같음을 분석한다.

• 대한전자공학회:학술대회논문집
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• 대한전자공학회 2000년도 ITC-CSCC -1
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• pp.302-305
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• 2000

This paper proposed an optimal register resource allocation algorithm using graph coloring for minimal register at high level synthesis. The proposed algorithm constructed interference graph consist of the intermediated representation CFG to description VHDL. and at interference graph fur the minimal select color selected a position node at stack, the next inserted spill code and the graph coloring process executes for optimal register allocation. The proposed algorithm proves to effect that result compare another allocation techniques through experiments of bench mark.

• 한국군사과학기술학회지
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• 제18권4호
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• pp.441-450
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• 2015

The tactical communications network has to be deployed rapidly at military operation area and support the communications between the military command systems and the weapon systems. For that, the frequency assignment is required for backbone wireless links of tactical communications network without frequency interferences. In this paper, we propose a frequency assignment method using net filter discrimination (NFD) and graph coloring to avoid frequency interferences. The proposed method presents frequency assignment problem of tactical communications network as vertex graph coloring problem of a weighted graph. And it makes frequency assignment sequences and assigns center frequencies to communication links according to the priority of communication links and graph coloring. The evaluation shows that this method can assign center frequencies to backbone communication links without frequency interferences. It also shows that the method can improve the frequency utilization in comparison with HTZ-warfare that is currently used by Korean Army.

• 한국컴퓨터정보학회논문지
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• 제24권10호
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• pp.57-64
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• 2019

이 논문에서는 인터벌 그래프 컬러링 역문제 중 다항시간 안에 풀이 가능한 경우에 대해 연구한다. 인터벌 그래프의 컬러링 역문제는 주어진 인터벌 그래프를 K개의 서로 다른 색깔로 색칠할 수 없는 경우를 가정하며, 다음과 같이 정의된다. 주어진 인터벌 그래프가 K개의 색깔을 이용해서 모두 칠해질 수 있도록 인터벌 그래프와 연관되어 있는 인터벌 시스템을 최소한으로 수정하는 문제이다. 인터벌 시스템에서 두 인터벌이 부분적으로라도 서로 겹쳐있는 구간이 있을 경우 두 인터벌에 해당하는 노드들이 엣지로 연결되어 있음을 의미하고, 따라서 이 경우에는 해당 노드들을 같은 색깔을 이용해 칠할 수 없다. 따라서 겹쳐져 있는 인터벌들을 이동시켜 해당 그래프의 chromatic number를 바꿀 수 있다. 본 논문에서는 인터벌의 길이가 모두 1 또는 2이며, 인터벌의 이동이 본래 위치 대비 오른쪽으로만 가능하다는 제한이 있는 경우에 대해 집중 탐구한다. 이 문제를 해결하는 다항시간 알고리즘으로 sorting과 선입선출 방식을 사용하는 2단계 알고리즘을 제안한다.

• Journal of Communications and Networks
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• 제15권3호
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• pp.321-328
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• 2013

In this paper, we present a technique for obtaining conflict-free schedules for real-time automation protocol for industrial Ethernet (RAPIEnet) switches. Mathematical model of the switch is obtained using graph theory. Initially network traffic entry and exit parts in a single RAPIEnet switch are identified, so that a bipartite conflict graph can be constructed. The obtained conflict graph is transformed to three kinds of matrices to be used as inputs for our simulation model, and selection of any of the matrix forms is application-specific. A greedy edge-coloring algorithm is used to schedule the network traffic and to solve the minimum coloring problem. After scheduling, empty slots are identified for forwarding the non real-time traffic of asynchronous devices. Finally, an algorithm for synchronizing the schedules of adjacent switches is proposed using edge-contraction and minors. All simulations were carried out using Matlab.

• 대한수학회보
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• 제61권1호
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• pp.263-271
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• 2024

A vertex coloring of a graph G is called injective if any two vertices with a common neighbor receive distinct colors. A graph G is injectively k-choosable if any list L of admissible colors on V (G) of size k allows an injective coloring 𝜑 such that 𝜑(v) ∈ L(v) whenever v ∈ V (G). The least k for which G is injectively k-choosable is denoted by χ^l_i(G). For a planar graph G, Bu et al. proved that χ^l_i(G) ≤ ∆ + 6 if girth g ≥ 5 and maximum degree ∆(G) ≥ 8. In this paper, we improve this result by showing that χ^l_i(G) ≤ ∆ + 6 for g ≥ 5 and arbitrary ∆(G).

• Journal of applied mathematics & informatics
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• 제42권2호
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• pp.353-366
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• 2024

A total coloring of a graph G is an assignment of colors to both the vertices and edges of G, such that no two adjacent or incident vertices and edges of G are assigned the same colors. In this paper, we have discussed the total coloring of M(T_n), M(D_n), M(DT_n), M(AT_n), M(DA(T_n)), M(Q_n), M(AQ_n) and also obtained the total chromatic number of M(T_n), M(D_n), M(DT_n), M(AT_n), M(DA(T_n)), M(Q_n), M(AQ_n).

• Kyungpook Mathematical Journal
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• 제56권2호
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• pp.611-624
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• 2016

This paper gives for every positive integer n an explicit construction of a graph G with fewer than $15{\log}^2n-{\frac{5}{2}}{\log}n+28$ vertices, such that there exists a nontrivial factoring of n if and only if G is 3-colorable.

• 한국경영과학회지
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• 제19권3호
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• pp.219-233
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• 1994

A fixed k-coloring problem is introduced and dealt with by efficient heuristic algorithms. It is shown that the problem can be transformed into the graph partitioning problem. Initial coloring and improving methods are proposed for problems with and with and without the size restriction. Algorithm Move, LEE and OEE are developed by modifying the Kernighan-Lin's two way uniform partitioning procedure. The use of global information in the selection of the node and the color set made the proposed algorithms superior to the existing method. The computational result also shows that the superiority does not sacrifice the time demand of the proposed algorithms.