• 충청수학회지
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• 제19권1호
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• pp.37-47
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• 2006

In this paper we characterize the graphs whose inserted graphs are Hamiltonian, and we study the relationship between Hamiltonian graphs and inserted graphs. Also we prove that if a connected graph G contains at least 3 vertices then inserted graph of the square of G is Hamiltonian and if G contains at least 3 edges then the square of inserted graph of G is Hamiltonian.

• 정보처리학회논문지A
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• 제13A권3호
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• pp.253-260
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• 2006

본 논문에서는 피라미드 그래프에서의 헤밀톤 사이클 특성을 분석한다. 사이클 확장 연산을 이용하여 사이클의 크기를 확대시켜 나가는 일련의 과정을 통하여 헤밀톤 사이클을 찾을 수 있는 제시된 알고리즘을 적용함으로써 임의의 높이 N인 피라미드 그래프 내에 길이 $(4^N-1)/3$인 헤밀톤 사이클이 존재함을 증명한다.

• 대한수학회보
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• 제59권1호
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• pp.119-139
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• 2022

This paper deals with the λ-labeling and L(2, 1)-coloring of simple graphs. A λ-labeling of a graph G is any labeling of the vertices of G with different labels such that any two adjacent vertices receive labels which differ at least two. Also an L(2, 1)-coloring of G is any labeling of the vertices of G such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial λ-labeling f is given in a graph G. A general question is whether f can be extended to a λ-labeling of G. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of G. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in L(2, 1)-coloring and λ-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph K_n☐K_n and the generation of λ-squares.

• 대한수학회논문집
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• 제11권4호
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• pp.1137-1145
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• 1996

A Steinhaus graph is a labelled graph whose adjacency matrix $A = (a_{i,j})$ has the Steinhaus property : $a_{i,j} + a{i,j+1} \equiv a_{i+1,j+1} (mod 2)$. We consider random Steinhaus graphs with n labelled vertices in which edges are chosen independently and with probability $\frac{1}{2}$. We prove that almost all Steinhaus graphs are Hamiltonian like as in random graph theory.

• 대한전자공학회:학술대회논문집
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• 대한전자공학회 2002년도 하계종합학술대회 논문집(3)
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• pp.63-66
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• 2002

In this paper, we consider the hamiltonian properties of m ${\times}$ n mesh networks with two wrap-around edges. We describe sufficient condition that at least two edges should be added to a mesh to make it hamiltonian-connected. We propose two graphs, M1(ｍ,n) and M2(ｍ,n). These are obtained by adding one and two edges respectively in the ｍ${\times}$ｎ mesh. We show the hamiltonian properties of M1(ｍ,n) and prove that M2(m, n) is hamiltonian-connected using the hamiltonian properties of M1(ｍ,n).

• Journal of applied mathematics & informatics
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• 제26권3_4호
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• pp.531-540
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• 2008

We discuss the decomposition problems on circuit graphs and triangular graphs, and show how they can be applied to obtain results on spanning trees or hamiltonian cycles. We also prove that every circuit graph containing no separating 3-cycles can be extended by adding new edges to a triangular graph containing no separating 3-cycles.

• International Journal of Computer Science & Network Security
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• 제21권8호
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• pp.17-27
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• 2021

Non-abelian group based cryptosystems are a latest research inspiration, since they offer better security due to their non-abelian properties. In this paper, we propose a novel approach to non-abelian group based public-key cryptographic protocols using semidirect products of finite groups. An intractable problem of determining automorphisms and generating elements of a group is introduced as the underlying mathematical problem for the suggested protocols. Then, we show that the difficult problem of determining paths and cycles of Cayley graphs including Hamiltonian paths and cycles could be reduced to this intractable problem. The applicability of Hamiltonian paths, and in fact any random path in Cayley graphs in the above cryptographic schemes and an application of the same concept to two previous cryptographic protocols based on a Generalized Discrete Logarithm Problem is discussed. Moreover, an alternative method of improving the security is also presented.

• Korean Journal of Mathematics
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• 제10권1호
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• pp.29-35
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• 2002

It is well known that a random graph $G_{1/2}(n)$ is Hamiltonian almost surely. In this paper, we show that every quasirandom graph $G(n)$ with minimum degree $(1+o(1))n/2$ is also Hamiltonian.

• 대한수학회보
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• 제53권4호
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• pp.1197-1211
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• 2016

Let R be a finite commutative ring with nonzero identity. We define ${\Gamma}(R)$ to be the graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u of R such that x + uy is a unit of R. This graph provides a refinement of the unit and unitary Cayley graphs. In this paper, basic properties of ${\Gamma}(R)$ are obtained and the vertex connectivity and the edge connectivity of ${\Gamma}(R)$ are given. Finally, by a constructive way, we determine when the graph ${\Gamma}(R)$ is Hamiltonian. As a consequence, we show that ${\Gamma}(R)$ has a perfect matching if and only if ${\mid}R{\mid}$ is an even number.

• Journal of Information Processing Systems
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• 제13권6호
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• pp.1554-1574
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• 2017

This paper introduces a new type of determining factor for Pseudo Random Strings (PRS). This classification depends upon a mathematical property called Finite Induction (FI). FI is similar to a Markov Model in that it presents a model of the sequence under consideration and determines the generating rules for this sequence. If these rules obey certain criteria, then we call the sequence generating these rules FI a PRS. We also consider the relationship of these kinds of PRS's to Good/deBruijn graphs and Linear Feedback Shift Registers (LFSR). We show that binary sequences from these special graphs have the FI property. We also show how such FI PRS's can be generated without consideration of the Hamiltonian cycles of the Good/deBruijn graphs. The FI PRS's also have maximum Shannon entropy, while sequences from LFSR's do not, nor are such sequences FI random.