Displaying similar documents to “Colouring polytopic partitions in ℝ d ”

An inequality concerning edges of minor weight in convex 3-polytopes

Igor Fabrici, Stanislav Jendrol' (1996)

Discussiones Mathematicae Graph Theory

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Let e i j be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is 20 e 3 , 3 + 25 e 3 , 4 + 16 e 3 , 5 + 10 e 3 , 6 + 6 [ 2 / 3 ] e 3 , 7 + 5 e 3 , 8 + 2 [ 1 / 2 ] e 3 , 9 + 2 e 3 , 10 + 16 [ 2 / 3 ] e 4 , 4 + 11 e 4 , 5 + 5 e 4 , 6 + 1 [ 2 / 3 ] e 4 , 7 + 5 [ 1 / 3 ] e 5 , 5 + 2 e 5 , 6 ≥ 120 ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

Finding H -partitions efficiently

Simone Dantas, Celina M. H. de Figueiredo, Sylvain Gravier, Sulamita Klein (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We study the concept of an H -partition of the vertex set of a graph G , which includes all vertex partitioning problems into four parts which we require to be nonempty with only external constraints according to the structure of a model graph H , with the exception of two cases, one that has already been classified as polynomial, and the other one remains unclassified. In the context of more general vertex-partition problems, the problems addressed in this paper have these properties:...

Generalization of the Zlámal condition for simplicial finite elements in ℝ d

Jan Brandts, Sergey Korotov, Michal Křížek (2011)

Applications of Mathematics

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The famous Zlámal’s minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in 2 d . In this paper we present and discuss its generalization to simplicial partitions in any space dimension.

On k-intersection edge colourings

Rahul Muthu, N. Narayanan, C.R. Subramanian (2009)

Discussiones Mathematicae Graph Theory

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We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χ’ₖ(G). Let fₖ be defined by f ₖ ( Δ ) = m a x G : Δ ( G ) = Δ χ ' ₖ ( G ) . We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.

The crossing number of the generalized Petersen graph P [ 3 k , k ]

Stanley Fiorini, John Baptist Gauci (2003)

Mathematica Bohemica

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Guy and Harary (1967) have shown that, for k ≥ 3 , the graph P [ 2 k , k ] is homeomorphic to the Möbius ladder M 2 k , so that its crossing number is one; it is well known that P [ 2 k , 2 ] is planar. Exoo, Harary and Kabell (1981) have shown hat the crossing number of P [ 2 k + 1 , 2 ] is three, for k ≥ 2 . Fiorini (1986) and Richter and Salazar (2002) have shown that P [ 9 , 3 ] has crossing number two and that P [ 3 k , 3 ] has crossing number k , provided k ≥ 4 . We extend this result by showing that P [ 3 k , k ] also has crossing number k for all k ≥ 4 .