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128
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 34 (11 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Defective coloring revisited
 Journal of Graph Theory
, 1997
"... A graph is (k; d)colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3 ..."
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Cited by 31 (0 self)
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A graph is (k; d)colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2) and (5, 1)colorable, and that a graph of genus γ is (γ=(d + 1) + 4; d)colorable, where γ is the maximum chromatic number of a graph embeddable on the surface of genus γ. It is shown that the (2; k)coloring, for k 1; and the (3, 1)coloring problems are NPcomplete even for planar graphs. In general graphs (k; d)coloring is NPcomplete for k 3; d 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented. c © 1997 John Wiley & Sons, Inc. 1.
Genus distributions for two class of graphs
 J. Combin. Theory Ser. B
, 1989
"... The set of orient able imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genusrespecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. These are the first two infinite classes of graphs for wh ..."
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Cited by 29 (20 self)
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The set of orient able imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genusrespecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their polygonal orientable imbed dings in the sphere.
Smarandache MultiSpace Theory
, 2011
"... Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countr ..."
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Cited by 22 (12 self)
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Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means the WORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds. We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multispace came into being by purely logic.
Chromatic Numbers of Quadrangulations on Closed Surfaces
"... It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N k has chromatic number at least 4 if G has a cycl ..."
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Cited by 19 (2 self)
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It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N k has chromatic number at least 4 if G has a cycle of odd length which cuts open N k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity.
Minors in Lifts of Graphs
 In SODA
, 2004
"... We study here lifts and random lifts of graphs, as defined in [1]. We consider the Hadwiger number # and the Hajos number # of # lifts of K , and analyze their extremal as well as their typical values (that is, for random lifts). When # = 2, we show that n, and random lifts achieve the lo ..."
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Cited by 15 (0 self)
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We study here lifts and random lifts of graphs, as defined in [1]. We consider the Hadwiger number # and the Hajos number # of # lifts of K , and analyze their extremal as well as their typical values (that is, for random lifts). When # = 2, we show that n, and random lifts achieve the lower bound (as n # #).
Bridges between Geometry and Graph Theory
 in Geometry at Work, C.A. Gorini, ed., MAA Notes 53
"... Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another ..."
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Cited by 15 (8 self)
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Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.
The Author
, 2005
"... This is the published version (version of record) of: Bennett, P. N. and Oppermann, W. 200603, Are nurses the key to the increased uptake of frequent nocturnal home haemodialysis in Australia?, Renal Society of Australasia journal, vol. 2, no. 1, pp. 2229. Zealand dialysis workforce, Renal Society ..."
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Cited by 14 (0 self)
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This is the published version (version of record) of: Bennett, P. N. and Oppermann, W. 200603, Are nurses the key to the increased uptake of frequent nocturnal home haemodialysis in Australia?, Renal Society of Australasia journal, vol. 2, no. 1, pp. 2229. Zealand dialysis workforce, Renal Society of Australasia, vol. 5, no. 3, pp. 147151.