• 대한수학회보
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• 제52권2호
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• pp.467-481
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• 2015

Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph G is related to the $Poincar\acute{e}$ polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multipartite graphs G.

• East Asian mathematical journal
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• 제30권3호
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• pp.311-319
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• 2014

It is shown that, for an even positive integer m with $m{\geq}4$ and arbitrary positive integer k and t, the complete multipartite graph $K_{km+1(2t)}$ can be decomposed into edge-disjoint gregarious m-cycles in such a way that the decomposition is circulant.

• East Asian mathematical journal
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• 제26권5호
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• pp.655-670
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• 2010

The complete multipartite graph $K_{n(m)}$ with n $ {\geq}$ 4 partite sets of size m is shown to have a decomposition into 4-cycles in such a way that vertices of each cycle belong to distinct partite sets of $K_{n(m)}$, if 4 divides the number of edges. Such cycles are called gregarious, and were introduced by Billington and Hoffman ([2]) and redefined in [3]. We independently came up with the result of [3] by using a difference set method, and improved the result so that the composition is circulant, in the sense that it is invariant under the cyclic permutation of partite sets. The composition is then used to construct gregarious 4-cycle decompositions when one partite set of the graph has different cardinality than that of others. Some results on joins of decomposable complete multipartite graphs are also presented.

• East Asian mathematical journal
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• 제33권1호
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• pp.67-74
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• 2017

Let k, m and t be positive integers with $m{\geq}4$ and even. It is shown that $K_{km+1(2t)}$ has a decomposition into gregarious m-cycles. Also, it is shown that $K_{km(2t)}$ has a decomposition into gregarious m-cycles if ${\frac{(m-1)^2+3}{4m}}$ < k. In this article, we make a remark that the decompositions can be circulant in the sense that it is preserved by the cyclic permutation of the partite sets, and we will exhibit it by examples.

• 대한수학회지
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• 제45권6호
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• pp.1623-1634
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• 2008

The complete multipartite graph $K_{n(2t)}$ having n partite sets of size 2t, with $n\;{\geq}\;6$ and $t\;{\geq}\;1$, is shown to have a decomposition into gregarious 6-cycles, that is, the cycles which have at most one vertex from any particular partite set. Complete sets of differences of numbers in ${\mathbb{Z}}_n$ are used to produce starter cycles and obtain other cycles by rotating the cycles around the n-gon of the partite sets.

• 한국통신학회논문지
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• 제40권9호
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• pp.1734-1740
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• 2015

본 논문은 분산 저장 시스템에서 기존의 부분접속수를 일반화한 개념인 결합 부분접속수를 소개하고, 결합 부분접속수($r_1$=2, $r_2$=3 or 4)를 만족하는 부호 설계 방식을 제안한다. 결합 부분접속수란 다양한 수의 노드 손실을 복구하기 위해 필요한 노드 수 집합을 의미한다. 제안된 방식은 완전다분할그래프를 사용하여 부호 설계를 단순화한다. 또한 제안된 방식으로 임의의 양의 정수 t에 대해 (2,t)-가용도를 갖는 이진 부분접속복구 부호를 설계할 수 있다. 즉, 1개 노드 손실 시 t개의 서로소인 복구 집합으로부터 각각 복구가 가능하며, 이때 각 복구 집합의 크기는 최대 2이다. 이러한 성질은 핫 데이터의 병렬처리를 가능하게 하므로 분산 저장 시스템에서 중요한 의미를 갖는다.

• 대한수학회보
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• 제56권4호
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• pp.1059-1075
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• 2019

Let G be a graph with no isolated vertex. A total dominating set, abbreviated TDS of G is a subset S of vertices of G such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a TDS of G. In this paper, we study the total domination number of central graphs. Indeed, we obtain some tight bounds for the total domination number of a central graph C(G) in terms of some invariants of the graph G. Also we characterize the total domination number of the central graph of some families of graphs such as path graphs, cycle graphs, wheel graphs, complete graphs and complete multipartite graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total domination number of central graphs.

• East Asian mathematical journal
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• 제29권3호
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• pp.337-347
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• 2013

For an even integer m at least 4 and any positive integer $t$, it is shown that the complete equipartite graph $K_{km(2t)}$ can be decomposed into edge-disjoint gregarious m-cycles for any positive integer ${\kappa}$ under the condition satisfying ${\frac{{(m-1)}^2+3}{4m}}$ < ${\kappa}$. Here it will be called a gregarious cycle if the cycle has at most one vertex from each partite set.

• Journal of applied mathematics & informatics
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• 제34권3_4호
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• pp.341-354
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• 2016

In this paper, the exact formulae for the generalized product degree distance, reciprocal product degree distance and product degree distance of tensor product of a connected graph and the complete multipartite graph with partite sets of sizes m₀, m₁, ⋯ , m_r−1 are obtained.

• 대한수학회보
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• 제44권4호
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• pp.709-719
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• 2007

In [8], it is shown that the complete multipartite graph $K_{n(2t)}$ having n partite sets of size 2t, where $n{\geq}6\;and\;t{\geq}1$, has a decomposition into gregarious 6-cycles if $n{\equiv}0,1,3$ or 4 (mod 6). Here, a cycle is called gregarious if it has at most one vertex from any particular partite set. In this paper, when $n{\equiv}0$ or 3 (mod 6), another method using difference set is presented. Furthermore, when $n{\equiv}0$ (mod 6), the decomposition obtained in this paper is ${\infty}-circular$, in the sense that it is invariant under the mapping which keeps the partite set which is indexed by ${\infty}$ fixed and permutes the remaining partite sets cyclically.