• Journal of applied mathematics & informatics
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• v.34 no.1_2
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• pp.49-60
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• 2016

In this paper, two different approaches to chromatic number of a bipolar fuzzy graph are introduced. The first approach is based on the α-cuts of a bipolar fuzzy graph and the second approach is based on the definition of Eslahchi and Onagh for chromatic number of a fuzzy graph. Finally, the bipolar fuzzy vertex chromatic number and the edge chromatic number of a complete bipolar fuzzy graph, characterized.

• Journal of applied mathematics & informatics
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• v.39 no.1_2
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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.

• International Journal of Computer Science & Network Security
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• v.22 no.9
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• pp.31-34
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• 2022

The concept of locating chromatic number of graph is a development of the concept of vertex coloring and partition dimension of graph. The locating-chromatic number of G, denoted by χ_L(G) is the smallest number such that G has a locating k-coloring. In this paper we will discussed about the procedure for determine the locating chromatic number of Origami graph using Python Programming.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.525-534
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• 2019

Let $P(G,{\lambda})$ denote the number of proper vertex colorings of G with ${\lambda}$ colors. The chromatic polynomial $P(C_n,{\lambda})$ for the cycle graph $C_n$ is well-known as $$P(C_n,{\lambda})=({\lambda}-1)^n+(-1)^n({\lambda}-1)$$ for all positive integers $n{\geq}1$. Also its inductive proof is widely well-known by the deletion-contraction recurrence. In this paper, we give this inductive proof again and three other proofs of this formula of the chromatic polynomial for the cycle graph $C_n$.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.435-440
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• 2006

Let G be a simple graph with vertex set V(G) and edge set E(G). A subset S of E(G) is called an edge cover of G if the subgraph induced by S is a spanning subgraph of G. The maximum number of edge covers which form a partition of E(G) is called edge covering chromatic number of G, denoted by X'c(G). It is known that for any graph G with minimum degree ${\delta},\;{\delta}-1{\le}X'c(G){\le}{\delta}$. If $X'c(G) ={\delta}$, then G is called a graph of CI class, otherwise G is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification of nearly bipartite graph and give some sufficient conditions for a nearly bipartite graph to be of CI class.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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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.

• 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.

• Journal of the Korea Society of Computer and Information
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• v.20 no.4
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• pp.111-117
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• 2015

The exam scheduling problem has been classified as nondeterministic polynomial time-complete (NP-complete) problem because of the polynomial time algorithm to obtain the exact solution has been unknown yet. Gu${\acute{e}}$ret et al. tries to obtain the solution using linear programming with $O(m^4)$ time complexity for this problem. On the other hand, this paper suggests chromatic number algorithm with O(m) time complexity. The proposed algorithm converts the original data to incompatibility matrix for modules and graph firstly. Then, this algorithm packs the minimum degree vertex (module) and not adjacent vertex to this vertex into the bin $B_i$ with color $C_i$ in order to exam within minimum time period and meet the incompatibility constraints. As a result of experiments, this algorithm reduces the $O(m^4)$ of linear programming to O(m) time complexity for exam scheduling problem, and gets the same solution with linear programming.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.15 no.1
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• pp.269-276
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• 2015

This paper proves the Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture of the vertex coloring problem, which is so far unresolved. The Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture states that "the union of k copies of k-cliques intersecting in at most one vertex pairwise is k-chromatic." i.e., x(G)=k. In a bid to prove this conjecture, this paper employs a method in which it determines the number of intersecting vertices and that of cliques that intersect at one vertex so as to count a vertex of the minimum degree ${\delta}(G)$ in the Minimum Independent Set (MIS) if both the numbers are even and to count a vertex of the maximum degree ${\Delta}(G)$ in otherwise. As a result of this algorithm, the number of MIS obtained is x(G)=k. When applied to $K_k$-clique sum intersecting graphs wherein $3{\leq}k{\leq}8$, the proposed method has proved to be successful in obtaining x(G)=k in all of them. To conclude, the Erd$\ddot{o}$s-Faber-Lov$\acute{a}$sz conjecture implying that "the k-number of $K_k$-clique sum intersecting graph is k-chromatic" is proven.

• Journal of the Korea Society of Computer and Information
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• v.16 no.6
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• pp.179-186
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• 2011

This paper suggests the minimum number of vertex guards algorithm. Given n rooms, the exact number of minimum vertex guards is proposed. However, only approximation algorithms are presented about the maximum number of vertex guards for polygon and orthogonal polygon without or with holes. Fisk suggests the maximum number of vertex guards for polygon with n vertices as follows. Firstly, you can triangulate with n-2 triangles. Secondly, 3-chromatic vertex coloring of every triangulation of a polygon. Thirdly, place guards at the vertices which have the minority color. This paper presents the minimum number of vertex guards using dominating set. Firstly, you can obtain the visibility graph which is connected all edges if two vertices can be visible each other. Secondly, you can obtain dominating set from visibility graph or visibility matrix. This algorithm applies various art galley problems. As a results, the proposed algorithm is simple and can be obtain the minimum number of vertex guards.