• The KIPS Transactions:PartA
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• v.18A no.6
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• pp.225-232
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• 2011

Restricted Hypercube-Like (RHL) graphs are a graph class that widely includes useful interconnection networks such as crossed cube, Mobius cube, Mcube, twisted cube, locally twisted cube, multiply twisted cube, and generalized twisted cube. In this paper, we show that for an m-dimensional RHL graph G, $m{\geq}4$, with an arbitrary faulty edge set $F{\subset}E(G)$, ${\mid}F{\mid}{\leq}m-2$, graph $G{\setminus}F$ has a hamiltonian path between any distinct two nodes s and t if dist(s, V(F))${\neq}1$ or dist(t, V(F))${\neq}1$. Graph $G{\setminus}F$ is the graph G whose faulty edges are removed. Set V(F) is the end vertex set of the edges in F and dist(v, V(F)) is the minimum distance between vertex v and the vertices in V(F).

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.629-646
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• 2004

Let G be a compact semialgebraic group and M a semi-algebraic G-set. We prove that there exists a semialgebraic slice at every point of M. Moreover M can be covered by finitely many semialgebraic G-tubes. As an application we give a different proof that every semialgebraic G-set admits a semi algebraic G-embedding into some semialgebraic orthogonal representation space of G, which has been proved in [15].

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1279-1300
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• 2021

Let P be a finite point set in ℝ² with the set of distance n-chains defined as ∆_n(P) = {(|p₁ - p₂|, |p₂ - p₃|, …, |p_n - p_n+1|) : p_i ∈ P}. We show that for 2 ⩽ n = O_|P|(1) we have ${\mid}{\Delta}_n(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^n}{{\log}^{\frac{13}{2}(n-1)}{\mid}P{\mid}}}$. Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in [28], which allows one to iterate efficiently on intersecting nested subsets of Guth-Katz lines. Let G is a simple connected graph on m = O(1) vertices with m ⩾ 2. Define the graph-distance set ∆_G(P) as ∆_G(P) = {(|p_i - p_j|)_{i,j}∈E(G) : p_i, p_j ∈ P}. Combining with results of Guth and Katz [17] and Rudnev [28] with the above, if G has a Hamiltonian path we have ${\mid}{\Delta}_G(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^{m-1}}{\text{polylog}{\mid}P{\mid}}}$.

• Journal of the Korean Mathematical Society
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• v.35 no.2
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• pp.371-386
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• 1998

Let G be a compact Lie group and M a semialgebraic G space in some orthogonal representation space of G. We prove that if G is finite then M has an equivariant semialgebraic triangulation. Moreover this triangulation is unique. When G is not finite we show that M has a semialgebraic G CW complex structure, and this structure is unique. As a consequence compact semialgebraic G space has an equivariant simple homotopy type.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.1
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• pp.13-23
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• 2022

Let G be a semialgebraic group not necessarily compact. Let M be a proper semialgebraic G-set whose orbit space has a semialgebraic structure. In this paper, we prove the embeddability of M into a G-representation space when G is linear.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.363-375
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• 2019

Let G = (V, E) be a simple graph with p vertices and q edges. A subset S of V (G) is called a strong (weak) efficient dominating set of G if for every $v{\in}V(G)$ we have ${\mid}N_s[v]{\cap}S{\mid}=1$ (resp. ${\mid}N_w[v]{\cap}S{\mid}=1$), where $N_s(v)=\{u{\in}V(G):uv{\in}E(G),\;deg(u){\geq}deg(v)\}$. The minimum cardinality of a strong (weak) efficient dominating set of G is called the strong (weak) efficient domination number of G and is denoted by ${\gamma}_{se}(G)$ (${\gamma}_{we}(G)$). A graph G is strong efficient if there exists a strong efficient dominating set of G. In this paper, some cycle and star related Nordhaus-Gaddum type relations on strong efficient dominating sets and the number of strong efficient dominating sets are studied.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.20 no.43
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• pp.153-162
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• 1997

M/G/1 queueing system is one of the most widely used one to model the real system. When operating a real systems, since it often takes cost, some control policies that change the operation scheme are adopted. In particular, the N-policy is the most popular among many control policies. Almost all researches on queueing system are based on the assumption that the arrival rates of customers into the queueing system is constant, In this paper, we consider the M/G/1 queueing system whose arrival rate varies according to the servers status : idle, set-up and busy states. For this study, we construct the steady state equations of queue lengths by means of the supplementary variable method, and derive the PGF(probability generating function) of them. The L-S-T(Laplace Stieltjes transform) of waiting time and average waiting time are also presented. We also develop an algorithm to find the optimal N-value from which the server starts his set-up. An analysis on the performance measures to minimize total operation cost of queueing system is included. We finally investigate the behavior of system operation cost as the optimal N and arrival rate change by a numerical study.

• Kyungpook Mathematical Journal
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• v.49 no.2
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• pp.283-288
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• 2009

Let N be a zero-symmetric near-ring with identity and let ${\Gamma}(N)$ be a graph with vertices as elements of N, where two different vertices a and b are adjacent if and only if + = N, where is the ideal of N generated by x. Let ${\Gamma}_1(N)$ be the subgraph of ${\Gamma}(N)$ generated by the set {n ${\in}$ N : = N} and ${\Gamma}_2(N)$ be the subgraph of ${\Gamma}(N)$ generated by the set $N{\backslash}{\upsilon}({\Gamma}_1(N))$, where ${\upsilon}(G)$ is the set of all vertices of a graph G. In this paper, we completely characterize the diameter of the subgraph ${\Gamma}_2(N)$ of ${\Gamma}(N)$. In addition, it is shown that for any near-ring, ${\Gamma}_2(N){\backslash}M(N)$ is a complete bipartite graph if and only if the number of maximal ideals of N is 2, where M(N) is the intersection of all maximal ideals of N and ${\Gamma}_2(N){\backslash}M(N)$ is the graph obtained by removing the elements of the set M(N) from the vertices set of the graph ${\Gamma}_2(N)$.