• 대한수학회지
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• 제49권3호
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• pp.475-491
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• 2012

In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of $I$-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each $I$-graph $I(n,j,k)$ admits a unit-distance representation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every $I$-graph $I(n,j,k)$ has an isomorphic $I$-graph that admits a unit-distance representation in the Euclidean plane with a $n$-fold rotational symmetry, with the exception of the families $I(n,j,j)$ and $I(12m,m,5m)$, $m{\geq}1$. We also provide unit-distance representations for these graphs.

• 한국정보과학회논문지:시스템및이론
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• 제35권6호
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• pp.263-272
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• 2008

본 논문은 피터슨 그래프를 기본으로 만든 새로운 상호연결망 피터슨-토러스(PT) 네트워크를 제안한다. PT 네트워크는 동일한 노드수를 갖는 honeycomb 토러스보다 짧은 지름을 갖고 망 비용이 개선된 연결망이다. 본 논문에서는 PT 네트워크의 최적 라우팅 알고리즘과 해밀톤 사이클 알고리즘을 제안하고, 지름, 망 비용 그리고 bisection width를 분석하였다.

• 자연과학논문집
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• 제12권1호
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• pp.1-9
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• 2002

연결 그래프의 꼭지점에 자갈이 분포되어 있다고 하자. 한 꼭지점에서 두 개의 자갈을 취하여 한 개의 자갈만을 인접한 꼭지점에 보내는 이동을 할 때, 자갈이 분포될 수 있는 모든 경우에서 임의의 꼭지점에 한 개의 자갈을 보내기 위해 필요한 최소의 자갈의 수를 그 그래프의 pebbling number 라고 한다. 이 논문에서 Petersen Graph의 pebbling 수를 계산하였고 complete bipartite 그래프 $K_{m,n}$과 꼭지점의 수 h가 4개 이상인 complete 그래프의 categorical product 의 pebbling number가 (m+n)h 이 됨을 보였다.

• Journal of Applied and Pure Mathematics
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• 제5권1_2호
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• pp.41-53
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• 2023

Let G = (V, E) be a (p, q) graph. Define $${\rho}=\{{\frac{2}{p}},\;{\text{{\qquad} if p is even}}\\{\frac{2}{p-1}},\;{{\text{if p is odd}}$$ and L = {±1, ±2, ±3, … , ±ρ} called the set of labels. Consider a mapping f : V ⟶ L by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that ${\mid}{\Delta}_{f_1}-{\Delta}_{f^c_1}{\mid}{\leq}1$, where ${\Delta}_{f_1}$ and ${\Delta}_{f^c_1}$ respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behaviour of Petersen graphs P(n, k) like P(n, 2), P(n, 3), P(n, 4).

• 대한수학회보
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• 제50권6호
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• pp.2021-2026
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• 2013

Let G = (V,E) be a graph. A function $f:V{\rightarrow}\{-1,+1\}$ defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. The signed total domination number of G, ${\gamma}^s_t(G)$, is the minimum weight of a signed total dominating function of G. In this paper, we study the signed total domination number of generalized Petersen graphs P(n, 2) and prove that for any integer $n{\geq}6$, ${\gamma}^s_t(P(n,2))=2[\frac{n}{3}]+2t$, where $t{\equiv}n(mod\;3)$ and $0 {\leq}t{\leq}2$.

• ETRI Journal
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• 제31권3호
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• pp.327-329
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• 2009

In a network, broadcasting is the dissemination of a message from a source node holding a message to all the remaining nodes through a call. This letter proposes a one-to-all broadcasting algorithm in the Petersen-torus network PT(n, n) for the single-link-available and multiple-link-available models. A PT(n, n) is a regular network whose degree is 4 and number of nodes is $10n^2$, where the Petersen graph is set as a basic module, and the basic module is connected in the form of a torus. A broadcasting algorithm is developed using a divide-and-conquer technique, and the time complexity of the proposed algorithm approximates n+4, the diameter of PT(n, n), which is the lower bound of the time complexity of broadcasting.

• 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.

• 한국정보과학회:학술대회논문집
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• 한국정보과학회 2002년도 가을 학술발표논문집 Vol.29 No.2 (3)
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• pp.298-300
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• 2002

급격한 기술의 발전 및 신기술의 등장에 따라 국가적 차원에서 고속, 대량의 데이터를 처리하는 네트워크를 구축할 필요성이 발생하였다. 이에 고속 통신망을 구축, 운영 중에 있으나, 현재의 network은 망의 안정성, 생존성 확보를 위하여 다수의 장거리 전용회선을 사용하고 있다. 본 논문에서는 현재의 network 구조에 피터슨 그래프를 이용하여 약간의 수정을 가하여 기존 운영중인 망에서 생존성을 보장하고 경제성을 향상시키는 효율적 망 활용 방법을 제시한다.