• 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 Korea Society of Computer and Information
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• v.18 no.11
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• pp.159-165
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• 2013

This paper proposes a O(E) polynomial-time algorithm that has been devised to simultaneously solve edge-coloring problem and graph classification problem both of which remain NP-complete. The proposed algorithm selects an edge connecting maximum and minimum degree vertices so as to determine the number of edge coloring ${\chi}^{\prime}(G)$. Determined ${\chi}^{\prime}(G)$ is in turn either ${\Delta}(G)$ or ${\Delta}(G)+1$. Eventually, the result could be classified as class 1 if ${\chi}^{\prime}(G)={\Delta}(G)$ and as category 2 if ${\chi}^{\prime}(G)={\Delta}(G)+1$. This paper also proves Vizing's planar graph conjecture, which states that 'all simple, planar graphs with maximum degree six or seven are of class one, closing the remaining possible case', which has known to be NP-complete.

• Journal of the Korea Society of Computer and Information
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• v.18 no.9
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• pp.121-129
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• 2013

This paper proposes a linear-time algorithm that has been designed to obtain an accurate solution for Dominating Set (DS) problem, which is known to be NP-complete due to the deficiency of polynomial-time algorithms that successfully derive an accurate solution to it. The proposed algorithm does so by repeatedly assigning vertex v with maximum degree ${\Delta}(G)$among vertices adjacent to the vertex v with minimum degree ${\delta}(G)$ to Minimum Independent DS (MIDS) as its element and removing all the incident edges until no edges remain in the graph. This algorithm finally transforms MIDS into Minimum DS (MDS) and again into Minimum Connected DS (MCDS) so as to obtain the accurate solution to all DS-related problems. When applied to ten different graphs, it has successfully obtained accurate solutions with linear time complexity O(n). It has therefore proven that Dominating Set problem is rather a P-problem.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.325-331
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• 2010

Let a and b be nonnegative integers with 2 $\leq$ a < b, and let G be a Hamiltonian graph of order n with n > $\frac{(a+b-5)(a+b-3)}{b-2}$. An [a, b]-factor F of G is called a Hamiltonian [a, b]-factor if F contains a Hamiltonian cycle. In this paper, it is proved that G has a Hamiltonian [a, b]-factor if $\delta(G)\;\geq\;\frac{(a-1)n+a+b-3)}{a+b-3}$ and $\delta(G)$ > $\frac{(a-2)n+2{\alpha}(G)-1)}{a+b-4}$.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.19 no.3
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• pp.193-199
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• 2019

In this paper I propose an algorithm of linear time complexity for NP-complete Maximum Independent Set (MIS) problem. Based on the basic property of the MIS, which forbids mutually adjoining vertices, the proposed algorithm derives the solution by repeatedly selecting vertices in the ascending order of their degree, given that the degree remains constant when vertices ${\nu}$ of the minimum degree ${\delta}(G)$ are selected and incidental edges deleted in a graph of n vertices. When applied to 22 graphs, the proposed algorithm could obtain the MIS visually yet effortlessly. The proposed linear MIS algorithm of time complexity O(n) always executes ${\alpha}(G)$ times, the cardinality of the MIS, and thus could be applied as a general algorithm to the MIS problem.

• 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 applied mathematics & informatics
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• v.26 no.5_6
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• pp.1085-1100
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• 2008

A set D of vertices of a graph G = (V(G),E(G)) is called a dominating set if every vertex of V(G) - D is adjacent to at least one element of D. The domination number of G, denoted by ${\gamma}(G)$, is the size of its smallest dominating set. Haynes et al.[5] present a conjecture: For any graph G with ${\delta}(G){\geq}k$,$\gamma(G){\leq}\frac{k}{3k-1}n$. When $k\;{\neq}\;6$, the conjecture was proved in [7], [8], [10], [12] and [13] respectively. In this paper we prove that every graph G on n vertices with ${\delta}(G)\;{\geq}\;6$ has a dominating set of order at most $\frac{6}{17}n$. Thus the conjecture was completely proved.

• Bulletin of the Korean Mathematical Society
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• v.48 no.2
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• pp.353-363
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• 2011

Let k $\geq$ 1 be an integer, and let G be a 2-connected graph of order n with n $\geq$ max{7, 4k+1}, and the minimum degree $\delta(G)$ $\geq$ k+1. In this paper, it is proved that G has a fractional k-factor excluding any given edge if G satisfies max{$d_G(x)$, $d_G(y)$} $\geq$ $\frac{n}{2}$ for each pair of nonadjacent vertices x, y of G. Furthermore, it is showed that the result in this paper is best possible in some sense.

• Journal of the Korea Society of Computer and Information
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• v.16 no.7
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• pp.85-93
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• 2011

The Vertex Coloring Problem hasn't been solved in polynomial time, so this problem has been known as NP-complete. This paper suggests linear time algorithm for Vertex Coloring Problem (VCP). The proposed algorithm is based on assumption that we can't know a priori the minimum chromatic number ${\chi}(G)$=k for graph G=(V,E) This algorithm divides Vertices V of graph into two parts as independent sets $\overline{C}$ and cover set C, then assigns the color to $\overline{C}$. The element of independent sets $\overline{C}$ is a vertex ${\upsilon}$ that has minimum degree ${\delta}(G)$ and the elements of cover set C are the vertices ${\upsilon}$ that is adjacent to ${\upsilon}$. The reduced graph is divided into independent sets $\overline{C}$ and cover set C again until no edge is in a cover set C. As a result of experiments, this algorithm finds the ${\chi}(G)$=k perfectly for 26 Graphs that shows the number of selecting ${\upsilon}$ is less than the number of vertices n.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.909-912
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• 2010

In this note we present a short proof of the following result by Zhou, Liu and Xu. Let G be a graph of order n, and let a and b be two integers with 1 $\leq$ a < b and b $\geq$ 3, and let g and f be two integer-valued functions defined on V(G) such that a $\leq$ g(x) < f(x) $\leq$ b for each $x\;{\in}\;V(G)$ and f(V(G)) - V(G) even. If $n\;{\geq}\;\frac{(a+b-1)^2+1}{a}$ and $\delta(G)\;{\geq}\;\frac{(b-1)n}{a+b-1}$,then G has a connected (g, f)-factor.