• Journal of Digital Convergence
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• v.20 no.2
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• pp.345-351
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• 2022

For simple undirected graph G=(V,E), L(2,1)-coloring of G is a nonnegative real-valued function f : V → [0,1,…,k] such that whenever vertices x and y are adjacent in G then |f(x)-f(y)|≥ 2 and whenever the distance between x and y is 2, |f(x)-f(y)|≥ 1. For a given L(2,1)-coloring c of graph G, the c-span is λ(c) = max{|c(v)-c(v)||u,v∈V}. L(2,1)-coloring number λ(G) = min{λ(c)} where the minimum is taken over all L(2,1)-coloring c of graph G. In this paper, based on Harary's Theorem, we use Tabu Search to figure out the existence of Hamiltonian Path in a complementary graph and confirmed that if λ(G) is equal to n(=|V|).

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.1_2
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• pp.86-94
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• 2004

In this paper, we show that $ G(2^m, 4), m{\geq}3$with at most m-2 faulty elements has a fault-free cycle of length 1 for every ${\leq}1{\leq}2^m-f_v$ is the number of faulty vertices. To achieve our purpose, we define a graph G to be k-fault hypohamiltonian-connected if for any set F of faulty elements, G- F has a fault-free path joining every pair of fault-free vertices whose length is shorter than a hamiltonian path by one, and then show that$ G(2^m, 4), m{\geq}3$ is m-3-fault hypohamiltonian-connected.

• Journal of KIISE:Computer Systems and Theory
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• v.30 no.12
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• pp.691-698
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• 2003

In this paper, we propose a problem, called one-to-one disjoint path cover problem, whether or not there exist k disjoint paths joining a pair of vertices which pass through all the vertices other than the two exactly once. A graph which for an arbitrary k, has a one-to-one disjoint path cover between an arbitrary pair of vertices has a hamiltonian property stronger than hamiltonian-connectedness. We investigate this problem on recursive circulants and prove that for an arbitrary k $k(1{\leq}k{\leq}m)$$ G(2^m,4)$,$m{\geq}3$, has a one-to-one disjoint path cover consisting of k paths between an arbitrary pair of vortices.

• The Transactions of the Korea Information Processing Society
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• v.4 no.11
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• pp.2701-2710
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• 1997

Recursive circulant graph has recently developed as a new model of multiprocessors, and drawn considerable attention to supercomputing, In this paper, we investigate the routing of a message i recursive circulant, that is a key to the performance of this network. On recursive circulant network, we would like to transmit m packets from a source node to a destination node simultaneously along paths, where the ith packet will traverse along the ith path $(o{\leq}i{\leq}m-1)$. In oder for all packets to arrive at the destination node quickly and securely, the ith path must be node-disjoint from all other paths. For construction of these paths, employing the Hamiltonian Circuit Latin Square(HCLS), a special class of $(n{\times}n)$ matrices, we present $O(n^2)$ parallel routing algorithm on recursive circulant network.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.21-38
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• 2006

Programmed grammars, one of the most important and well investigated classes of grammars with context-free rules and a mechanism controlling the application of the rules, can be described by graphs. We investigate whether or not the restriction to special classes of graphs restricts the generative power of programmed grammars with erasing rules and without appearance checking, too. We obtain that Eulerian, Hamiltonian, planar and bipartite graphs and regular graphs of degree at least three are pr-universal in that sense that any language which can be generated by programmed grammars (with erasing rules and without appearance checking) can be obtained by programmed grammars where the underlying graph belongs to the given special class of graphs, whereas complete graphs, regular graphs of degree 2 and backbone graphs lead to proper subfamilies of the family of programmed languages.

• 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 Magnetic Resonance Society
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• v.17 no.2
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• pp.92-97
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• 2013

We study the electron spin resonance line-width (ESRLW) of $Fe^{3+}$ in crystalline $LiNbO_3$ ; the ESRLW is obtained using the projection operator method (POM) developed by Argyres and Sigel. The ESRLW is calculated to be axially symmetric about the c-axis and is analyzed by the spin Hamiltonian with an isotopic g factor at a frequency of 9.5 GHz. The ESRLW increases exponentially as the temperature increases, and the ESRLW is almost constant in the high-temperature region (T>8000 K). This kind of temperature dependence of the ESRLW indicates a motional narrowing of the spectrum when $Fe^{3+}$ ions substitute the $Nb^{5+}$ ions in an off-center position. It is clear from this feature that there are two different regions in the graph of the temperature dependence of the ESRLW.