Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

Volkmann, Lutz,Winzen, Stefan Department of Mathematics 2008 Kyungpook mathematical journal Vol.48 No.2

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.

SIGNED STAR k-DOMINATION AND k-DOMATIC NUMBERS OF DIGRAPHS

S. M. Sheikholeslami,L. Volkmann 장전수학회 2015 Advanced Studies in Contemporary Mathematics Vol.25 No.3

Let D be a simple digraph with vertex set V (D) and arc set A(D), and let k be a positive integer. A function f , A(D) →{-1, 1} is said to be a signed star k-dominating function (SSkD function) on D if ∑a∈A(v) f(a) ≥ k for every vertex v of D, where A(v) is the set of arcs with head v. The signed star k-domination number of a digraph D is defined by γkSS(D) = min{∑a∈2A(D) f(a) | f is a SSkD function on D}. A set {f1, f2, ... , fd} of distinct signed star k-dominating functions on D with the property that ∑d i=1 fi(a) ≤| 1 for each arc a∈A(D), is called a signed star k-dominating family (of functions) on D. The maximum number of functions in a signed star k-dominating family on D is the signed star k-domatic number of D, denoted by dkSS(D). In this paper we study properties of the signed star k-domination and k-domatic number dkSS(D) of digraphs D. Some of our results extend these one given by Atapour, Sheikholeslami, Ghameshlou and Volkmann [1] for the the signed star domatic number, Sheikholeslami and Volkmann [5] for the the signed star k-domatic number and Xu [8] for the signed star domination number of a graph.

CYCLES THROUGH A GIVEN SET OF VERTICES IN REGULAR MULTIPARTITE TOURNAMENTS

Volkmann, Lutz,Winzen, Stefan Korean Mathematical Society 2007 대한수학회지 Vol.44 No.3

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 $r{\geq}g(c)$.

Complementary Cycles in Regular Multipartite Tournaments, Where One Cycle Has Length Four

Lutz Volkmann 경북대학교 자연과학대학 수학과 2004 Kyungpook mathematical journal Vol.44 No.2

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.In 1999, Yeo conjectured that each regular c-partite tournament D with c 4 and jV(D)j 8 contains a pair of vertex disjoint directed cycles of lengths 4 and jV(D)j 4. An example will demonstrate that Yeo’s conjecture is not true in general for regular 4-partite tournaments with two vertices in each partite. However, in all other cases we shall confirm this conjecture in affirmative.

Verfassungsänderung und Verfassungswandel-Beobachtungen zum Verhältnis von Stabilität und Dynamik im Verfassungsrecht der Bundesrepublik Deutschland-

Uwe Volkmann 헌법재판연구원 2018 헌법재판연구 Vol.5 No.1

Im Rechtsvergleich lassen sich ganz allgemein stärker dynamische (flexible, offene etc.) von stärker statischen (festen, rigiden etc.) Verfassungen unterscheiden, wobei damit nur die äußersten Pole einer gleitenden Skala bezeichnet sind. Für die nähere Zuordnung einer konkreten Verfassung zu einem dieser Pole darf dabei nicht nur die Möglichkeit der formellen Verfassungsänderung in der Verfassung selbst eingerichteten Verfahren in den Blick genommen werden, sondern mindestens ebenso sehr auch die Möglichkeit der informellen Verfassungsänderung durch Interpretation, die in der Verfassungstheorie oft auch zu einer eigenständigen Kategorie des „Verfassungswandels“ zusammengezogen wird. Das Grundgesetz erweist sich danach im internationalen Vergleich als eine Verfassung von ausgesprochen hoher Dynamik: Einerseits haben sich die Hürden für formelle Verfassungsänderungen (im Wesentlichen eine Zweidrittelmehrheit in beiden Kammern) als nicht besonders hoch erwiesen, so dass es in seiner kurzen Geschichte relativ häufig geändert worden ist. Andererseits wird das Grundgesetz namentlich vom Bundesverfassungsgericht sehr stark nach dem Muster einer „living constitution“ interpretiert; dabei sind vor allem die Grundrechte kontinuierlich an Veränderungen des gesellschaftlichen Umfelds angepasst und auf neue Herausforderungen eingestellt worden. Der Vortrag stellt zunächst die beiden Formen der Verfassungsentwicklung einander gegenüber, bevor in einem letzten Schritt gefragt wird, wie sich dies auf den Stellenwert der Verfassung im öffentlichen Leben der Bundesrepublik ausgewirkt hat.

Cycles through a given set of vertices in regular multipartite tournaments

Lutz Volkmann,Stefan Winzen 대한수학회 2007 대한수학회지 Vol.44 No.3

A tournament is an orientation of a complete graph, and in generala multipartite or c-partite tournament is an orientation of acomplete c-partite graph.In a recent article, the authors proved that a regular c-partitetournament with r ge 2 vertices in each partite set contains acycle with exactly r-1 vertices from each partite set, withexception of the case that c = 4 and r = 2. Here we willexamine the existence of cycles with r-2 vertices from eachpartite set in regular multipartite tournaments where the r-2vertices are chosen arbitrarily. Let D be a regular c-partitetournament and let X subseteq V(D) be an arbitrary set withexactly 2 vertices of each partite set. For all c ge 4 wewill determine the minimal value g(c) such that D-X isHamiltonian for every regular multipartite tournament with r geg(c).

ROMAN k-DOMINATION IN GRAPHS

Kammerling, Karsten,Volkmann, Lutz Korean Mathematical Society 2009 대한수학회지 Vol.46 No.6

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) $\rightarrow$ {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices $\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$ with $f(\upsilon_i)$ = 2 for i = 1, 2, $\ldot$, k. The weight of a Roman k-dominating function is the value f(V (G)) = $\sum_{u{\in}v(G)}$ f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number ${\gamma}_{kR}$(G) of G. Note that the Roman 1-domination number $\gamma_{1R}$(G) is the usual Roman domination number $\gamma_R$(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

The k-Rainbow Domination and Domatic Numbers of Digraphs

Sheikholeslami, S.M.,Volkmann, Lutz Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.1

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set $\{1,2,{\ldots},k\}$ such that for any vertex $v{\in}V(D)$ with $f(v)={\emptyset}$ the condition ${\cup}_{u{\in}N^-(v)}$ $f(u)=\{1,2,{\ldots},k\}$ is fulfilled, where $N^-(v)$ is the set of in-neighbors of v. A set $\{f_1,f_2,{\ldots},f_d\}$ of k-rainbow dominating functions on D with the property that $\sum_{i=1}^{d}{\mid}f_i(v){\mid}{\leq}k$ for each $v{\in}V(D)$, is called a k-rainbow dominating family (of functions) on D. The maximum number of functions in a k-rainbow dominating family on D is the k-rainbow domatic number of D, denoted by $d_{rk}(D)$. In this paper we initiate the study of the k-rainbow domatic number in digraphs, and we present some bounds for $d_{rk}(D)$.

Roman k-domination in graphs

Karsten Kammerling,Lutz Volkmann 대한수학회 2009 대한수학회지 Vol.46 No.6

Let k be a positive integer, and let G be a simple graph with vertex set V(G). A Roman k-dominating function on G is a function f:V(G)→{0,1,2} such that every vertex u for which f(u)=0 is adjacent to at least k vertices v_1,v_2,…,v_k with f(v_i)=2 for i=1,2,…,k. The weight of a Roman k-dominating function is the value f(V(G))=∑{u∈V(G)}f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number γ_{kR}(G) of G. Note that the Roman 1-domination number γ_{1R}(G) is the usual Roman domination number γ_R(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number. Let k be a positive integer, and let G be a simple graph with vertex set V(G). A Roman k-dominating function on G is a function f:V(G)→{0,1,2} such that every vertex u for which f(u)=0 is adjacent to at least k vertices v_1,v_2,…,v_k with f(v_i)=2 for i=1,2,…,k. The weight of a Roman k-dominating function is the value f(V(G))=∑{u∈V(G)}f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number γ_{kR}(G) of G. Note that the Roman 1-domination number γ_{1R}(G) is the usual Roman domination number γ_R(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Modification of Biophysical Soil Properties by Biochar Amendments

Gerardo Ojeda,Joana Patricio,Stefania Mattanna,Anna Avila,Martin Volkmann,Josep Maria Alcaniz,Jorg Bachmann 한국토양비료학회 2014 한국토양비료학회 학술발표회 초록집 Vol.2014 No.6