• Kyungpook Mathematical Journal
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• 제60권2호
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• pp.349-359
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• 2020

We study perfect 2-colorings of regular graphs. In particular, we consider the 4-regular case. We obtain a characterization of perfect 2-colorings of toroidal grids.

• 대한수학회보
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• 제55권5호
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• pp.1397-1404
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• 2018

For a connected graph G, an r-maximum edge-coloring is an edge-coloring f defined on E(G) such that at every vertex v with $d_G(v){\geq}r$ exactly r incident edges to v receive the maximum color. The r-maximum index $x^{\prime}_r(G)$ is the least number of required colors to have an r-maximum edge coloring of G. In this paper, we show how the r-maximum index is affected by adding an edge or a vertex. As a main result, we show that for each $r{\geq}3$ the r-maximum index function over the graphs admitting an r-maximum edge-coloring is unbounded and the range is the set of natural numbers. In other words, for each $r{\geq}3$ and $k{\geq}1$ there is a family of graphs G(r, k) with $x^{\prime}_r(G(r,k))=k$. Also, we construct a family of graphs not admitting an r-maximum edge-coloring with arbitrary maximum degrees: for any fixed $r{\geq}3$, there is an infinite family of graphs ${\mathcal{F}}_r=\{G_k:k{\geq}r+1\}$, where for each $k{\geq}r+1$ there is no r-maximum edge-coloring of $G_k$ and ${\Delta}(G_k)=k$.

• 대한수학회보
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• 제49권2호
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• pp.359-365
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• 2012

Giving a planar graph G, let $x^'_l(G)$ and $x^{''}_l(G)$ denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if a planar graph G without 6-cycles with chord, then $x^'_l(G){\leq}{\Delta}(G)+1$ and $x^{''}_l(G){\leq}{\Delta}(G)+2$ where ${\Delta}(G){\geq}6$.

• 한국컴퓨터정보학회논문지
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• 제21권3호
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• pp.1-8
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• 2016

The printed circuit board (PCB) can be used only 2 layers of front and back. Therefore, the wiring line segments are located in 2 layers without crossing each other. In this case, the line segment can be appear in both layers and this line segment is to resolve the crossing problem go through the via. The via minimization problem (VMP) has minimum number of via in layout design problem. The VMP is classified by NP-complete because of the polynomial time algorithm to solve the optimal solution has been unknown yet. This paper suggests polynomial time algorithm that can be solve the optimal solution of VMP. This algorithm transforms n-line segments into vertices, and p-crossing into edges of a graph. Then this graph is partitioned into 3-coloring sets of each vertex in each set independent each other. For 3-coloring sets $C_i$, (i=1,2,3), the $C_1$ is assigned to front F, $C_2$ is back B, and $C_3$ is B-F and connected with via. For the various experimental data, though this algorithm can be require O(np) polynomial time, we obtain the optimal solution for all of data.

• 한국정보통신학회논문지
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• 제9권5호
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• pp.1102-1110
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• 2005

본 논문에서는 기존 복셀 칼라링 기법에서의 색상 일관성에 대한 문턱값 문제를 해결하기 위한 개선된 복셀 칼라링 기법을 제안하였다. 제안 기법에서는 표면 복셀에 대한 색상 일관성의 문턱값을 내부 복셀의 색상 일관성 값으로 대체함으로써 복셀 칼라링의 반복 회수가 증가함에 따라 개별 표면 복셀에 대한 최적의 문턱값을 찾아가도록 하였다. 또한 그래프 절단 기법을 적용하여 주위 복셀을 제거 판단에 함께 고려함으로써 표면 잡음을 감소시켰다.

• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.511-524
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• 2023

An L(p₁, p₂, p₃, . . . , p_m)-labeling of a graph G is an assignment of non-negative integers, called as labels, to the vertices such that the vertices at distance i should have at least pi as their label difference. If p₁ = 4, p₂ = 3, p₃ = 2, p₄ = 1, then it is called a L(4, 3, 2, 1)-labeling which is widely studied in the literature. A L(4, 3, 2, 1)-path coloring of graphs, is a labeling g : V (G) → Z⁺ such that there exists at least one path P between every pair of vertices in which the labeling restricted to this path is a L(4, 3, 2, 1)-labeling. This concept was defined and results for some simple graphs were obtained by the same authors in an earlier article. In this article, we study the concept of L(4, 3, 2, 1)-path coloring for complete bipartite graphs, 2-edge connected split graph, Cartesian product and join of two graphs and prove an existence theorem for the same.

• 한국경영과학회지
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• 제18권3호
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• pp.35-49
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• 1993

The node coloring problem is a problem to color the nodes of a graph using the minimum number of colors possible so that any two adjacent nodes are colored differently. This problem, along with the edge coloring problem, has a variety of practical applications particularly in item loading, resource allocation, exam timetabling, and channel assignment. The node coloring problem is an NP-hard problem, and thus many researchers develop a number of heuristic algorithms. In this paper, we survey and classify those heuristics with the emphasis on how an algorithm orders the nodes and colors the nodes using a determined ordering.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2001년도 ICCAS
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• pp.49.3-49
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• 2001

In 1994 Adelman´s pioneer work demonstrated that deoxyribonucleic acid (DNA) could be used as a medium for computation to solve mathematical problems. He described the use of DNA based computational approach to solve the Hamiltonian Path Problem (HPP). Since then a number of combinatorial problems have been analyzed by DNA computation approaches including, for example: Maximum Independent Set (MIS), Maximal Clique and Satisfaction (SAT) Problems. In the present paper we propose a method of solving another classic combinatorial optimization problem - the eraph Coloring Problem (GCP), using specifically designed circular DNA plasmids as a computation tool. The task of the analysis is to color the graph so that no two nodes ...

• 대한전자공학회:학술대회논문집
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• 대한전자공학회 2002년도 ITC-CSCC -3
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• pp.1375-1377
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• 2002

New graph coloring problems are discussed as models of a multihop network in this report. We consider a total scheduling problem, and prove that this problem is NP-hard. We propose new scheduling models of a multi-hop network for CDMA system, and show the complexity results of the scheduling problems.

• 대한수학회지
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• 제48권1호
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• pp.105-115
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• 2011

Let f be a function which assigns a positive integer f(v) to each vertex v $\in$ V (G), let r, s and t be non-negative integers. An f-coloring of G is an edge-coloring of G such that each vertex v $\in$ V (G) has at most f(v) incident edges colored with the same color. The minimum number of colors needed to f-color G is called the f-chromatic index of G and denoted by ${\chi}'_f$(G). An [r, s, t; f]-coloring of a graph G is a mapping c from V(G) $\bigcup$ E(G) to the color set C = {0, 1, $\ldots$; k - 1} such that |c($v_i$) - c($v_j$ )| $\geq$ r for every two adjacent vertices $v_i$ and $v_j$, |c($e_i$ - c($e_j$)| $\geq$ s and ${\alpha}(v_i)$ $\leq$ f($v_i$) for all $v_i$ $\in$ V (G), ${\alpha}$ $\in$ C where ${\alpha}(v_i)$ denotes the number of ${\alpha}$-edges incident with the vertex $v_i$ and $e_i$, $e_j$ are edges which are incident with $v_i$ but colored with different colors, |c($e_i$)-c($v_j$)| $\geq$ t for all pairs of incident vertices and edges. The minimum k such that G has an [r, s, t; f]-coloring with k colors is defined as the [r, s, t; f]-chromatic number and denoted by ${\chi}_{r,s,t;f}$ (G). In this paper, we present some general bounds for [r, s, t; f]-coloring firstly. After that, we obtain some important properties under the restriction min{r, s, t} = 0 or min{r, s, t} = 1. Finally, we present some problems for further research.