• Title/Summary/Keyword: multipartite

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CYCLES THROUGH A GIVEN SET OF VERTICES IN REGULAR MULTIPARTITE TOURNAMENTS

  • Volkmann, Lutz;Winzen, Stefan
    • Journal of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.683-695
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    • 2007
  • A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with $r{\geq}2$ vertices in each partite set contains a cycle with exactly r-1 vertices from each partite set, with exception of the case that c=4 and r=2. Here we will examine the existence of cycles with r-2 vertices from each partite set in regular multipartite tournaments where the r-2 vertices are chosen arbitrarily. Let D be a regular c-partite tournament and let $X{\subseteq}V(D)$ be an arbitrary set with exactly 2 vertices of each partite set. For all $c{\geq}4$ we will determine the minimal value g(c) such that D-X is Hamiltonian for every regular multipartite tournament with $r{\geq}g(c)$.

Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

  • Volkmann, Lutz;Winzen, Stefan
    • Kyungpook Mathematical Journal
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    • v.48 no.2
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    • pp.287-302
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    • 2008
  • The vertex set of a digraph D is denoted by V (D). A c-partite tournament is an orientation of a complete c-partite graph. A digraph D is called cycle complementary if there exist two vertex disjoint cycles $C_1$ and $C_2$ such that V(D) = $V(C_1)\;{\cup}\;V(C_2)$, and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles $C_1$ and $C_2$ such that $V(C_1)\;{\cup}\;V(C_2)$ contains vertices of all partite sets of D. The problem of complementary cycles in 2-connected tournaments was completely solved by Reid [4] in 1985 and Z. Song [5] in 1993. They proved that every 2-connected tournament T on at least 8 vertices has complementary cycles of length t and ${\mid}V(T)\mid$ - t for all $3\;{\leq}\;t\;{\leq}\;{\mid}V(T)\mid/2$. Recently, Volkmann [8] proved that each regular multipartite tournament D of order ${\mid}V(D)\mid\;\geq\;8$ is cycle complementary. In this article, we analyze multipartite tournaments that are weakly cycle complementary. Especially, we will characterize all 3-connected c-partite tournaments with $c\;\geq\;3$ that are weakly cycle complementary.

v-PATHS OF ARCS IN REGULAR MULTIPARTITE TOURNAMENTS

  • Yao, Tianxing;Guo, Yubao;Zhang, Kemin
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.389-394
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    • 1999
  • A v-path of an arc xy in a multipartite tournament T is an oriented oath in T-y which starts at x such that y does not dominate and end vertex of the path. We show that if T is a regular n-partite (n$\geq$7) tournament, then every arc of T has a v-path of length m for all m satisfying 2$\leq$m$\leq$n-2. Our result extends the corresponding result for regular tournaments, due to Alspach, Reid and Roselle [2] in 1974, to regular multipartite tournaments.

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DECOMPOSITIONS OF COMPLETE MULTIPARTITE GRAPHS INTO GREGARIOUS 6-CYCLES USING COMPLETE DIFFERENCES

  • Cho, Jung-R.;Gould, Ronald J.
    • Journal of the Korean Mathematical Society
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    • v.45 no.6
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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.

A DIFFERENCE SET METHOD FOR CIRCULANT DECOMPOSITIONS OF COMPLETE PARTITE GRAPHS INTO GREGARIOUS 4-CYCLES

  • Kim, Eun-Kyung;Cho, Young-Min;Cho, Jung-Rae
    • East Asian mathematical journal
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    • v.26 no.5
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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.

A REMARK ON CIRCULANT DECOMPOSITIONS OF COMPLETE MULTIPARTITE GRAPHS BY GREGARIOUS CYCLES

  • Cho, Jung Rae
    • East Asian mathematical journal
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    • v.33 no.1
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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.

SIGNED A-POLYNOMIALS OF GRAPHS AND POINCARÉ POLYNOMIALS OF REAL TORIC MANIFOLDS

  • Seo, Seunghyun;Shin, Heesung
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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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.

Concurrence of Rank-two Multipartite Quantum States (2-계수 양자상태의 양자얽힘 분석)

  • Bae, Joonwoo
    • Korean Journal of Optics and Photonics
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    • v.29 no.2
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    • pp.64-69
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    • 2018
  • In this work, we derive a general lower bound to concurrence of an arbitrary mixture of two pure states, that is, rank-two multipartite quantum states. We show that the lower bound can tightly detect entanglement of rank-two states, and also can be implemented experimentally with present-day technologies, i.e. single-copy level measurement and classical post-processing.

A NOTE ON DECOMPOSITION OF COMPLETE EQUIPARTITE GRAPHS INTO GREGARIOUS 6-CYCLES

  • Cho, Jung-Rae
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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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.