• 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.

• Kyungpook Mathematical Journal
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• 제59권4호
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• pp.591-601
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• 2019

Let ℤ_n be the ring of integers modulo n and Γ(ℤ_n) the zero-divisor graph of ℤ_n. In this paper, we study some properties of Γ(ℤ_n). More precisely, we completely characterize the diameter and the girth of Γ(ℤ_n). We also calculate the chromatic number of Γ(ℤ_n).

• 대한수학회보
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• 제53권1호
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• pp.139-151
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• 2016

A k-total-coloring of a graph G is a coloring of $V{\cup}E$ using k colors such that no two adjacent or incident elements receive the same color. The total chromatic number ${\chi}^{{\prime}{\prime}}(G)$ of G is the smallest integer k such that G has a k-total-coloring. Let G be a planar graph with maximum degree ${\Delta}$. In this paper, it's proved that if ${\Delta}{\geq}7$ and G does not contain adjacent 5-cycles, then the total chromatic number ${\chi}^{{\prime}{\prime}}(G)$ is ${\Delta}+1$.

• Kyungpook Mathematical Journal
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• 제60권4호
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• pp.723-729
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• 2020

Let ℤ_n be the ring of integers modulo n and let ℤ_n[X]] be either ℤ_n[X] or ℤ_n[[X]]. Let 𝚪(Z_n[X]]) be the zero-divisor graph of ℤ_n[X]]. In this paper, we study some properties of 𝚪(ℤ_n[X]]). More precisely, we completely characterize the diameter and the girth of 𝚪(ℤ_n[X]]). We also calculate the chromatic number of 𝚪(ℤ_n[X]]).

• Journal of applied mathematics & informatics
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• 제40권3_4호
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• pp.729-740
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• 2022

Let ℤ be the ring of integers and let ℤ_n be the ring of integers modulo n. Let ℤ(+)ℤ_n be the idealization of ℤ_n in ℤ and let (ℤ(+)ℤ_n)[X]] be either (ℤ(+)ℤ_n)[X] or (ℤ(+)ℤ_n)[[X]]. In this article, we study the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. More precisely, we completely characterize the diameter and the girth of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. We also calculate the chromatic number of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]].

• 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).

• 한국컴퓨터정보학회논문지
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• 제20권4호
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• pp.111-117
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• 2015

시험 일정 계획 수립 문제는 정확한 해를 다항시간으로 구하는 알고리즘이 알려져 있지 않은 NP-완전이다. 이 문제에 대해, Gu${\acute{e}}$ret et al.은 $O(m^4)$ 수행 복잡도의 선형계획법으로 해를 얻고자 하였다. 반면에, 본 논문에서는 O(m) 복잡도의 채색 수 알고리즘을 제안하였다. 제안된 방법은 원 데이터를 교과목에 대한 부적합성 행렬과 그래프로 변환시켰다. 다음으로, 부적합성 제약조건을 충족하면서 최소의 시간으로 시험을 치루기 위해, 최소 차수 정점(교과목)부터 인접하지 않은 정점들을 $C_i$ 색으로 배정하여 $B_i$ 상자에 채웠다. 실험 결과, 제안된 알고리즘은 시험 일정 계획 수립 문제에 대해 선형계획법의 $O(m^4)$를 O(m)으로 단축시키면서도 동일한 해를 얻었다.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제11권2호
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• pp.650-669
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• 2017

Coordinated multiple points transmission and reception (CoMP) technology is used to mitigate the inter-cell interference, and increase cell average user normalized throughput and cell edge user normalized throughput. There are two kinds of radio resource schedule strategies in LTE-A/5G CoMP system, and they are called centralized scheduling strategy and distributed scheduling strategy. The regional centralized scheduling cannot solve interference of inter-region, and the distributed scheduling leads to worse efficiency in the utilize of resources. In this paper, a novel distributed scheduling scheme named 9-Cell alternate authorization (9-CAA) is proposed. In our scheme, time-domain resources are divided orthogonally by coloring theory for inter-region cooperation in 9-Cell scenario [6]. Then, we provide a formula based on 0-1 integer programming to get chromatic number in 9-CAA. Moreover, a feasible optimal chromatic number search algorithm named CNS-9CAA is proposed. In addition, this scheme is expanded to 3-Cell scenario, and name it 3-Cell alternate authorization (3-CAA). At last, simulation results indicate that 9/3-CAA scheme exceed All CU CoMP, 9/3C CU CoMP and DLC resource scheduling scheme in cell average user normalized throughput. Especially, compared with the non-CoMP scheme as a benchmark, the 9-CAA and 3-CAA have improved the edge user normalized throughput by 17.2% and 13.0% respectively.

• 대한수학회논문집
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• 제38권3호
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• pp.937-941
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• 2023

For each positive integer n, a square congruence graph S(n) is the graph with vertex set H = {1, 2, 3,...., n} and two vertices a, b are adjacent if they are distinct and a² ≡ b² (mod n). In this paper we investigate some structural properties of square congruence graph and we obtain the relationship between clique number, chromatic number and maximum degree of square congruence graph. Also we study square congruence graph with p vertices or 2p vertices for any prime number p.

• 한국연초학회지
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• 제26권1호
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• pp.35-42
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• 2004

This study was carried out to examinate the effect of aging period on chromatic, chemical and organoleptic characteristics, and to evaluate of optimum aging period for each grade in burley leaf tobacco. The leaf tobaccos were produced in 2000, and threshed, redried and packed in carton box under the current methods. Four grades of processed leaf(A3T, B1T, C1W and D3W) were stored during 24 months (May 10, 2001 to April 31, 2003) in warehouse of Chungju Leaf Tobacco Processing Factory. The leaf tobaccos were sampled at three month intervals for analysis of chromatic, chemical and organoleptic properties. pH values of four grades, and Yellow(b) of A3T and BIT were also significantly decreased during the aging. Filling values, tar, nicotine and CO contents of tobacco smoke, and puff number of cigarettes were not significantly changed during the aging. Positive correlation coefficients were significantly observed between irritation of the calculated attributes from contents of volatile oil components in leaf tobacco and that of the panel sensory attributes. The ratio of maximum change in irritation attribute was larger than that in taste attribute during aging. The calculated irritations of 18～24 months aging for A3T and B1T were significantly lower than zero-aging, while those of C1W and D3W were not significant among aging periods. The panel irritations of 12～15 months aging for four grades were low tendency. The results suggest that decrease of aging period for thin leaf from 21～24 to 12～15 months may be beneficial to save storage cost in burley tobacco.