• Journal of applied mathematics & informatics
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• 제19권1_2호
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• pp.127-133
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• 2005

An f-coloring of a graph G = (V, E) is a coloring of edge set E such that each color appears at each vertex v $\in$ V at most f(v) times. The minimum number of colors needed to f-color G is called the f-chromatic index $\chi'_f(G)$ of G. Any graph G has f-chromatic index equal to ${\Delta}_f(G)\;or\;{\Delta}_f(G)+1,\;where\;{\Delta}_f(G)\;=\;max\{{\lceil}\frac{d(v)}{f(v)}{\rceil}\}$. If $\chi'_f(G)$= ${\Delta}$f(G), then G is of $C_f$ 1 ; otherwise G is of $C_f$ 2. In this paper, the classification problem of complete graphs on f-coloring is solved completely.

• 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.

• 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.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.357-366
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• 2007

Let G(V, E) be a simple graph, and let f be an integer function on V with $1{\leq}f(v){\leq}d(v)$ to each vertex $v{\in}V$. An f-edge cover-coloring of a graph G is a coloring of edge set E such that each color appears at each vertex $v{\in}V$ at least f(v) times. The f-edge cover chromatic index of G, denoted by ${\chi}'_{fc}(G)$, is the maximum number of colors such that an f-edge cover-coloring of G exists. Any simple graph G has an f-edge cover chromatic index equal to ${\delta}_f\;or\;{\delta}_f-1,\;where\;{\delta}_f{=}^{min}_{v{\in}V}\{\lfloor\frac{d(v)}{f(v)}\rfloor\}$. Let G be a connected and not complete graph with ${\chi}'_{fc}(G)={\delta}_f-1$, if for each $u,\;v{\in}V\;and\;e=uv{\nin}E$, we have ${\chi}'_{fc}(G+e)>{\chi}'_{fc}(G)$, then G is called an f-edge covered critical graph. In this paper, some properties on f-edge covered critical graph are discussed. It is proved that if G is an f-edge covered critical graph, then for each $u,\;v{\in}V\;and\;e=uv{\nin}E$ there exists $w{\in}\{u,v\}\;with\;d(w)\leq{\delta}_f(f(w)+1)-2$ such that w is adjacent to at least $d(w)-{\delta}_f+1$ vertices which are all ${\delta}_f-vertex$ in G.

• 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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• 제59권1호
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• pp.119-139
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• 2022

This paper deals with the λ-labeling and L(2, 1)-coloring of simple graphs. A λ-labeling of a graph G is any labeling of the vertices of G with different labels such that any two adjacent vertices receive labels which differ at least two. Also an L(2, 1)-coloring of G is any labeling of the vertices of G such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial λ-labeling f is given in a graph G. A general question is whether f can be extended to a λ-labeling of G. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of G. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in L(2, 1)-coloring and λ-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph K_n☐K_n and the generation of λ-squares.

• 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.

• 분석과학
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• 제26권1호
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• pp.99-105
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• 2013

염모제품 중 유효성분의 품질검사는 식품의약품안전청의 "의약외품에 관한 기준 및 시험방법"에 따라 제품에 표기된 산화염료들을 박층크로마토그래프법(TLC방법)으로 확인시험을 하도록 되어있다. 그러나 TLC방법은 원료분량이 미량이거나, $R_f$ 값이 비슷한 성분들이 존재하면 확인시험이 어려운 문제점이 있다. 본 연구에서는 미량성분 검출이 용이하며 분석시간이 짧다고 보고된 UPLC를 이용하기 위한 시료 전처리 및 분석조건을 탐색하였다. 유효성분들을 분석할 수 있는 검출한계는 6.7-77.9 ${\mu}g/L$, 정량한계는 22.3-259.7 ${\mu}g/L$ 이었으며, 회수율은 ${\alpha}$-naphthol를 제외하고는 96.2-101.5%로 양호하였다. 유효성분 추출하기 위한 시료 전처리에는 헥산-증류수를 사용하였다. 시료 전처리 시 복잡한 추출과정을 거치지 않고, UPLC방법은 빠르고 정확하게 염모제품에 함유되어 있는 유효성분들을 동시에 분석 할 수 있었다.

• 한국의류학회지
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• 제39권4호
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• pp.601-612
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• 2015

This research investigated the dyeing behavior of palmatine (a major coloring compound of Phellodendron bark in addition to berberine) using mercerization (M), tannic acid (T), mercerization-tannic acid (MT), and tannic acid -mercerization (TM) pretreatments. Mercerization was conducted using $20^{\circ}C$ of 20% NaOH for 5 minutes. Tannic acid treatment was conducted using 15% o.w.f. solution of tannic acid at $60^{\circ}C$ for 30 minutes and fixed with potassium antimonyl tartrate trihydrate. Dyeing was conducted using 1% o.w.f. palmatine chloride hydrate with 1:100 liquor ratio at $10-95^{\circ}C$ for 10-60 minutes in a dyebath of pH 3-9. MT method resulted in the highest dye uptake and was two times higher than the average dye uptake of T method. MT method provided the best result when the dyeing temperature was $80^{\circ}C$ or $95^{\circ}C$ and the dyeing time was 60 minutes. The pH of the dyebath had less effect on the dye uptake but a pH higher than 5 provided better results. The study confirmed that palmatine is a major coloring compound of Phellodendron bark and that the MT method can be used as a successful cotton dyeing method.

• 한국염색가공학회지
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• 제12권4호
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• pp.239-247
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• 2000

The coloring matter was extracted from the heartwood of Amur cork tree by distilled water. Change of UV-Visible spectra of coloring matter solution by extraction condition and stability for irradiation were determined, and the effect of repeated dyeing with condition of dyebath and mordanting method on shade depth and lightfastness were also investigated. The results are as follows : 1) Absorbance of Amur cork tree extract increased with the lapse of extraction time. λmax of color solution extracted from Amur cork tree was found at 420, 333, and 262nm. 2) Absorbance of Amur cork tree extract decreased remarkably after 2hr irradiation. 3) The K/S of silk fabrics increased with the increase of dyeing temperature, time, amounts of Amur cork tree for extraction, and pH of color solution. 4) K/S of silk fabrics dyed by repeated dyeing method was affected by pH and concentration of color solution. 5) Lightfastness of silk twice dyed with Amur cork tree extract after pre-mordanted by 8%(o.w.f) chromium acetate was moderately improved.