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## Best Possible (best + possible)
## Selected Abstracts## Non-rainbow colorings of 3-, 4- and 5-connected plane graphs JOURNAL OF GRAPH THEORY, Issue 2 2010k DvoAbstract We study vertex-colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3 -connected plane graph with n vertices, then the number of colors in such a coloring does not exceed . If G is 4 -connected, then the number of colors is at most , and for n,3(mod8), it is at most . Finally, if G is 5 -connected, then the number of colors is at most . The bounds for 3 -connected and 4 -connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 129,145, 2010 [source] ## On the maximum number of cycles in a planar graph JOURNAL OF GRAPH THEORY, Issue 3 2008R. E. L. AldredAbstract Let G be a graph on p vertices with q edges and let r,=,q,,,p,=,1. We show that G has at most cycles. We also show that if G is planar, then G has at most 2r,,,1,=,o(2r,,,1) cycles. The planar result is best possible in the sense that any prism, that is, the Cartesian product of a cycle and a path with one edge, has more than 2r,,,1 cycles. © Wiley Periodicals, Inc. J. Graph Theory 57: 255,264, 2008 [source] ## Maximum acyclic and fragmented sets in regular graphs JOURNAL OF GRAPH THEORY, Issue 2 2008Penny HaxellAbstract We show that a typical d -regular graph G of order n does not contain an induced forest with around vertices, when n,,,d,,,1, this bound being best possible because of a result of Frieze and ,uczak [6]. We then deduce an affirmative answer to an open question of Edwards and Farr (see [4]) about fragmentability, which concerns large subgraphs with components of bounded size. An alternative, direct answer to the question is also given. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 149,156, 2008 [source] ## Forcing highly connected subgraphs JOURNAL OF GRAPH THEORY, Issue 4 2007Maya Jakobine SteinAbstract A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex-degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex-degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2k at the vertices and a vertex-degree of order k log k at the ends which have no k -connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge-degree for the ends (which is defined as the maximum number of edge-disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge-degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)- edge -connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331,349, 2007 [source] ## Sufficient conditions for graphs to be ,,-optimal, super-edge-connected, and maximally edge-connected JOURNAL OF GRAPH THEORY, Issue 3 2005Angelika HellwigAbstract The restricted-edge-connectivity of a graph G, denoted by ,,(G), is defined as the minimum cardinality over all edge-cuts S of G, where G - S contains no isolated vertices. The graph G is called ,,-optimal, if ,,(G),=,,(G), where ,(G) is the minimum edge-degree in G. A graph is super-edge-connected, if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle-free, and bipartite graphs to be ,,-optimal, as well as conditions depending on the clique number. These conditions imply super-edge-connectivity, if , (G),,,3, and the equality of edge-connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228,246, 2005 [source] ## Toughness, minimum degree, and spanning cubic subgraphs JOURNAL OF GRAPH THEORY, Issue 2 2004D. BauerAbstract Degree conditions on the vertices of a t -tough graph G (1,,,t,<,3) are presented which ensure the existence of a spanning cubic subgraph in G. These conditions are best possible to within a small additive constant for every fixed rational t ,[1,4/3),[2,8/3). © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 119,141, 2004 [source] ## Nowhere-zero 3-flows in locally connected graphs JOURNAL OF GRAPH THEORY, Issue 3 2003Hong-Jian LaiAbstract Let G be a graph. For each vertex v ,V(G), Nv denotes the subgraph induces by the vertices adjacent to v in G. The graph G is locally k -edge-connected if for each vertex v ,V(G), Nv is k -edge-connected. In this paper we study the existence of nowhere-zero 3-flows in locally k -edge-connected graphs. In particular, we show that every 2-edge-connected, locally 3-edge-connected graph admits a nowhere-zero 3-flow. This result is best possible in the sense that there exists an infinite family of 2-edge-connected, locally 2-edge-connected graphs each of which does not have a 3-NZF. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211,219, 2003 [source] ## Degree sequence conditions for equal edge-connectivity and minimum degree, depending on the clique number JOURNAL OF GRAPH THEORY, Issue 3 2003Lutz VolkmannAbstract Using the well-known Theorem of Turán, we present in this paper degree sequence conditions for the equality of edge-connectivity and minimum degree, depending on the clique number of a graph. Different examples will show that these conditions are best possible and independent of all the known results in this area. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 234,245, 2003 [source] ## Girth of sparse graphs JOURNAL OF GRAPH THEORY, Issue 3 2002Béla BollobásAbstract For each fixed k,,,0, we give an upper bound for the girth of a graph of order n and size n,+,k. This bound is likely to be essentially best possible as n,,,,. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 194,200, 2002; DOI 10.1002/jgt.10023 [source] ## The times they are a changin JOURNAL OF NURSING MANAGEMENT, Issue 5 2009Cert Ed, MIKE THOMAS PhDAim, A discussion paper outlining the potential for a multi-qualified health practitioner who has undertaken a programme of study incorporating the strengths of the specialist nurse with other professional routes. Background and rationale, The concept and the context of ,nursing' is wide and generalized across the healthcare spectrum with a huge number of practitioners in separate branches, specialities and sub-specialities. As a profession, nursing consists of different groups in alliance with each other. How different is the work of the mental health forensic expert from an acute interventionalist, or a nurse therapist, from a clinical expert in neurological deterioration? The alliance holds because of the way nurses are educated and culturalized into the profession, and the influence of the statutory bodies and the context of a historical nationalized health system. This paper discusses the potential for a new type of healthcare professional, one which pushes the intra- and inter-professional agenda towards multi-qualified staff who would be able to work across current care boundaries and be more flexible regarding future care delivery. In September 2003, the Nursing and Midwifery Council stated that there were ,more than 656 000 practitioners' on its register and proposed that from April 2004, there were new entry descriptors. Identifying such large numbers of practitioners across a wide range of specialities brings several areas of the profession into question. Above all else, it highlights how nursing has fought and gained recognition for specialisms and that through this, it may be argued client groups receive the best possible ,fit' for their needs, wants and demands. However, it also highlights deficits in certain disciplines of care, for example, in mental health and learning disabilities. We argue that a practitioner holding different professional qualifications would be in a position to provide a more holistic service to the client. Is there then a gap for a ,new breed' of practitioner; ,a hybrid' that can achieve a balanced care provision to reduce the stress of multiple visits and multiple explanations? Methods, Review of the literature but essentially informed by the authors personal vision relating to the future of health practitioner education. Implications for nursing management, This article is of significance for nurse managers as the future workforce and skill mix of both acute and community settings will be strongly influenced by the initial preregistration nurse education. [source] ## Minimizing makespan on a single batch processing machine with nonidentical job sizes NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 3 2001Guochuan ZhangAbstract We deal with the problem of minimizing makespan on a single batch processing machine. In this problem, each job has both processing time and size (capacity requirement). The batch processing machine can process a number of jobs simultaneously as long as the total size of these jobs being processed does not exceed the machine capacity. The processing time of a batch is just the processing time of the longest job in the batch. An approximation algorithm with worst-case ratio 3/2 is given for the version where the processing times of large jobs (with sizes greater than 1/2) are not less than those of small jobs (with sizes not greater than 1/2). This result is the best possible unless P = NP. For the general case, we propose an approximation algorithm with worst-case ratio 7/4. A number of heuristics by Uzosy are also analyzed and compared. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 226,240, 2001 [source] ## On-line algorithms for minimizing makespan on batch processing machines NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 3 2001Gouchuan ZhangAbstract We consider problem of scheduling jobs on-line on batch processing machines with dynamic job arrivals to minimize makespan. A batch machine can handle up to B jobs simultaneously. The jobs that are processed together from a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is given by the longest processing time of any job in the batch. Each job becomes available at its arrival time, which is unknown in advance, and its processing time becomes known upon its arrival. In the first part of this paper, we address the single batch processing machine scheduling problem. First we deal with two variants: the unbounded model where B is sufficiently large and the bounded model where jobs have two distinct arrival times. For both variants, we provide on-line algorithms with worst-case ratio (the inverse of the Golden ratio) and prove that these results are the best possible. Furthermore, we generalize our algorithms to the general case and show a worst-case ratio of 2. We then consider the unbounded case for parallel batch processing machine scheduling. Lower bound are given, and two on-line algorithms are presented. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 241,258, 2001 [source] ## Minors in random regular graphs RANDOM STRUCTURES AND ALGORITHMS, Issue 4 2009Nikolaos FountoulakisAbstract We show that there is a constant c so that for fixed r , 3 a.a.s. an r -regular graph on n vertices contains a complete graph on vertices as a minor. This confirms a conjecture of Markström (Ars Combinatoria 70 (2004) 289,295). Since any minor of an r -regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph Gn,p during the phase transition (i.e., when pn , 1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source] ## Induced subgraphs with distinct sizes RANDOM STRUCTURES AND ALGORITHMS, Issue 1 2009Noga AlonAbstract We show that for every 0 < , < 1/2, there is an n0 = n0(,) such that if n > n0 then every n -vertex graph G of size at least and at most contains induced k -vertex subgraphs with at least 10,7k different sizes, for every . This is best possible, up to a constant factor. This is also a step toward a conjecture by Erd,s, Faudree, and Sós on the number of distinct pairs (|V (H)|,|E(H)|) of induced subgraphs of Ramsey graphs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source] ## Experimente mit Ozon: Angewandte Chemilumineszenz CHEMIE IN UNSERER ZEIT (CHIUZ), Issue 1 2007Steffen Albrecht PD Dr. habil.Es werden drei sehr gut sichtbare experimentell erprobte Chemilumineszenz-Demonstrationen bei Einsatz von Ozon als Oxidationsmittel vorgestellt. Die Experimente können auch in größeren Räumen bei nur mäßiger Verdunklung eindrucksvoll demonstriert werden. Es handelt sich um die Umsetzung des Ozons mit einem Phenazinfarbstoff, einem Luminophor und erstmals mit Ruthenium-organyl-Komplexen. Neben den Versuchsbeschreibungen wird soweit möglich auf den Reaktionsmechanismus eingegangen. Three impressive and well tried demonstrations of chemiluminescence in the use of ozone as oxidant are shown. The experiments can be presented even in larger rooms with only slight darkening. The mechanism is the reaction of ozone with a phenazine dye, a luminophor and , for the first time , with ruthenium-organyl-complexes. Additional to the description of the experiments the mechanism of the reaction are described best possible and same historic and theoretic aspects are covered as well. [source] |