• 대한수학회논문집
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• 제38권3호
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• pp.937-941
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• 2023

For each positive integer n, a square congruence graph S(n) is the graph with vertex set H = {1, 2, 3,...., n} and two vertices a, b are adjacent if they are distinct and a² ≡ b² (mod n). In this paper we investigate some structural properties of square congruence graph and we obtain the relationship between clique number, chromatic number and maximum degree of square congruence graph. Also we study square congruence graph with p vertices or 2p vertices for any prime number p.

• Korean Journal of Mathematics
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• 제30권2호
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• pp.391-402
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• 2022

For a given graph G = (V, E), a dominating set is a subset V' of the vertex set V so that each vertex in V ＼ V' is adjacent to a vertex in V'. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ(G). For an integer k ≥ 1, the k-th power G^k of a graph G with V (G^k) = V (G) for which uv ∈ E(G^k) if and only if 1 ≤ d_G(u, v) ≤ k. Note that G² is the square graph of a graph G. In this paper, we obtain some tight bounds for the sum of the domination numbers of a graph and its square graph in terms of the order, order and size, and maximum degree of the graph G. Also, we characterize such extremal graphs.

• 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.

• 대한수학회지
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• 제54권4호
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• pp.1121-1148
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• 2017

Go is an ancient game of great complexity and has a huge following in East Asia. It is also very rich mathematically, and can be played on any graph, although it is usually played on a square lattice. As with any game, one of the most fundamental problems is to determine the number of legal positions, or the probability that a random position is legal. A random Go position is generated using a model previously studied by the author, with each vertex being independently Black, White or Uncoloured with probabilities q, q, 1 - 2q respectively. In this paper we consider the probability of legality for two scenarios. Firstly, for an $N{\times}N$ square lattice graph, we show that, with $q=cN^{-{\alpha}}$ and c and ${\alpha}$ constant, as $N{\rightarrow}{\infty}$ the limiting probability of legality is 0, exp($-2c^5$), and 1 according as ${\alpha}$ < 2/5, ${\alpha}=2/5$ and ${\alpha}$ > 2/5 respectively. On the way, we investigate the behaviour of the number of captured chains (or chromons). Secondly, for a random graph on n vertices with edge probability p generated according to the classical $Gilbert-Erd{\ddot{o}}s-R{\acute{e}}nyi$ model ${\mathcal{G}}$(n; p), we classify the main situations according to their asymptotic almost sure legality or illegality. Our results draw on a variety of probabilistic and enumerative methods including linearity of expectation, second moment method, factorial moments, polyomino enumeration, giant components in random graphs, and typicality of random structures. We conclude with suggestions for further work.

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 2000년도 춘계학술대회 논문집
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• pp.739-742
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• 2000

From the experimental study of W-EDM for alloyed steel, the characteristics such as Hand Drum Form and surface roughness have been observed and evaluated for various conditions. In square hole, the increase of If as to made condition the calculate high value of surface roughness. Also compare dimensionless square hole with circle hole' graph. In circle hole, if a value of surface roughness IP 6 in a side of circle it show a 0.4${\mu}{\textrm}{m}$ and in IP 8, 0.6${\mu}{\textrm}{m}$, in IP 10, 0.7${\mu}{\textrm}{m}$, in IP 12. 0.8${\mu}{\textrm}{m}$ higher than before. This figure show the surface roughness is higher than before, because a table move in either X-axis or Y-axis in square hole, on the contrary, in circle there table move in X-axis and Y-axis at the same time. hand drum form getting small when wire tension increase 1000gf to 1500gf, at the same working conditions. the smaller of off time, the mailer of hand drum form in same condition and same wire tension. but if you compare square hole with circle hole' graph hand drum form displayed in maintained term of working condision, on the contrary, in case of square hole variation of hand drum form is more increase than a grow of IP

• 한국공작기계학회:학술대회논문집
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• 한국공작기계학회 2000년도 춘계학술대회논문집 - 한국공작기계학회
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• pp.567-572
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• 2000

From the experimental study of W-EDM for alloyed steel, the characteristics such as Hand Drum Form and surface roughness have been observed and evaluated for various conditions. In square hole, the increase of IP as to made condition, the calculate high value of surface roughness. Also compare dimensionless square hole with circle hole' graph, In circle hole, if a value of surface roughness IP 6 in a side of circle it show a 0.4${\mu}{\textrm}{m}$ and in IP 8, 0.6${\mu}{\textrm}{m}$, in IP 10, 0.7${\mu}{\textrm}{m}$, in IP 12, 0.8${\mu}{\textrm}{m}$ higher than before. This figure show the surface roughness is higher than before, because a table move in either X-axis or Y-axis in square hole, on the contrary, in circle there table move in X-axis and Y-axis at the same time. hand drum form getting small when wire tension increase 1000gf to 1500gf, at the same working conditions. The smaller of off time, the maller of hand drum form in same condition and same wire tension. but if you compare square hole with circle hole' graph, hand drum form displayed in maintained term of working condision, on the contrary, in case of square hole variation of hand drum form is more increase than a grow of IP

• 대한수학회보
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• 제55권6호
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• pp.1667-1690
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• 2018

We consider the minimum order of a graph G with a given lazy cop number $c_L(G)$. Sullivan, Townsend and Werzanski [7] showed that the minimum order of a connected graph with lazy cop number 3 is 9 and $k_3{\square}k_3$ is the unique graph on nine vertices which requires three lazy cops. They conjectured that for a graph G on n vertices with ${\Delta}(G){\geq}n-k^2$, $c_L(G){\leq}k$. We proved that the conjecture is true for k = 4. Furthermore, we showed that the Petersen graph is the unique connected graph G on 10 vertices with ${\Delta}(G){\leq}3$ having lazy cop number 3 and the minimum order of a connected graph with lazy cop number 4 is 16.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.39-48
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• 2006

Let ${\gamma}$(G) be the domination number of a graph G. It is shown that for any $k {\ge} 0$ there exists a Cartesian graph bundle $B{\Box}_{\varphi}F$ such that ${\gamma}(B{\Box}_{\varphi}F) ={\gamma}(B){\gamma}(F)-2k$. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality ${\gamma}(B{\bigotimes}_{\varphi}F){\le}{\gamma}(B){\gamma}(F)$ for strong graph bundles. Examples of graphs Band F with ${\gamma}(B{\bigotimes}_{\varphi}F) < {\gamma}(B){\gamma}(F)$ are given.

• 한국콘텐츠학회논문지
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• 제9권12호
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• pp.582-591
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• 2009

피라미드 그래프는 정방형 메쉬와 트리 구조를 기반으로 하는 상호연결망 토폴로지이다. 본 논문에서는 피라미드 그래프의 각 계층을 구성하고 있는 기반 그래프로서의 정방형 메쉬 그래프의 간선들을 두개의 서로 다른 그룹으로 분류하는 전략을 채택한다. 메쉬 내의 간선 집합은 해당 간선의 양 끝 정점들에 인접된 부모 정점들이 상위 계층 내에서 서로 이웃하는 관계인지 아니면 공유하는 관계인지에 따라서 각각 NPC-간선과 SPC-간선이라는 이름으로 불리는 두 개의 서로 다른 부분집합으로 나누어질 수 있다. 아울러 원래 그래프에서의 SPC-간선들을 압축된 결과 그래프에서는 압축된 슈퍼-정점 내부로 숨김으로써 NPC-간선들에만 초점을 맞출 수 있도록 하기 위해 압축 그래프의 개념을 소개한다. 본 논문에서는 $2^n\times2^n$ 2-차원 정방형 메쉬 내에서 헤밀톤 사이클 구성 시 포함할 수 있는 NPC-간선 개수의 하한 및 상한이 각각 $2^{2n-2}$와 $3*(2^{2n-2}-2^{n-1})$임을 분석한다. 이 결과를 피라미드 그래프로 확장시킴으로써 n-차원 피라미드 내에서 헤밀톤 사이클에 포함가능한 NPC-간선의 최대 개수가 $4^{n-1}-3*2^{n-1}$-2n+7 임을 증명한다.

• 한국멀티미디어학회논문지
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• 제13권8호
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• pp.1183-1193
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• 2010

피라미드 그래프는 병렬처리 분야에서 정방형 메쉬와 트리 구조를 기반으로 하는 상호연결망 위상으로 잘 알려져 있다. 개선된 피라미드 그래프는 이러한 피라미드 그래프보다 성능을 향상시키기 위해 메쉬를 토러스로 대체시킨 구조를 말한다. 본 논문에서는 개선된 피라미드 그래프의 각 계층을 형성하고 있는 기반 부-그래프로서의 정방형 토러스 그래프의 간선들을 두 개의 서로 다른 그룹으로 분류하는 전략을 채택한다. 토러스 그래프 내의 간선 집합은 해당 간선의 양 끝 정점들에 인접된 부모 정점들이 상위 계층에서 서로 인접하는지 아니면 공유하는 관계 인지에 따라 각각 NPC-간선과 SPC-간선이라 불리는 두 개의 서로 다른 부분집합으로 나누어 고려한다. 아울러 원래 그래프에서의 SPC-간선들을 압축된 결과 그래프에서는 압축된 슈퍼-정점 내부로 은닉시킴으로써 NPC-간선들에 대해서만 초점을 맞추도록 하기 위해 압축 그래프의 개념을 소개한다. 본 연구에서는 $2^n{\times}2^n$ 2-차원 정방형 토러스 내에서 헤밀톤 사이클 구성 시 포함할 수 있는 NPC-간선 개수의 하한 및 상한이 각각 $2^{2n-2}$와 $3{\cdot}2^{2n-2}$임을 분석한다. 이 결과를 개선된 피라미드 그래프로 확장시킴으로써 개선된 n-차원 피라미드 그래프 내에서 헤밀톤 사이클에 포함할 수 있는 NPC-간선의 최대 개수는 $4^{n-1}$-2n+1 개임을 증명한다.