8,620 research outputs found
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
End Graph Effects on Chromatic Polynomials for Strip Graphs of Lattices and their Asymptotic Limits
We report exact calculations of the ground state degeneracy per site
(exponent of the ground state entropy) of the -state Potts
antiferromagnet on infinitely long strips with specified end graphs for free
boundary conditions in the longitudinal direction and free and periodic
boundary conditions in the transverse direction. This is equivalent to
calculating the chromatic polynomials and their asymptotic limits for these
graphs. Making the generalization from to , we determine the full locus on which is nonanalytic in the
complex plane. We report the first example for this class of strip graphs
in which encloses regions even for planar end graphs. The bulk of
the specific strip graph that exhibits this property is a part of the Archimedean lattice.Comment: 27 pages, Revtex, 11 encapsulated postscript figures, Physica A, in
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