879 research outputs found
Global cycle properties in graphs with large minimum clustering coefficient
The clustering coefficient of a vertex in a graph is the proportion of
neighbours of the vertex that are adjacent. The minimum clustering coefficient
of a graph is the smallest clustering coefficient taken over all vertices. A
complete structural characterization of those locally connected graphs, with
minimum clustering coefficient 1/2 and maximum degree at most 6, that are fully
cycle extendable is given in terms of strongly induced subgraphs with given
attachment sets. Moreover, it is shown that all locally connected graphs with
minimum clustering coefficient 1/2 and maximum degree at most 6 are weakly
pancyclic, thereby proving Ryjacek's conjecture for this class of locally
connected graphs.Comment: 16 pages, two figure
Maximum Orders of Cyclic and Abelian Extendable Actions on Surfaces
Let be a closed surface embedded in . If a group
can acts on the pair , then we call such a group action on
extendable over .
In this paper we show that the maximum order of extendable cyclic group
actions is when is even and when is odd; the maximum
order of extendable abelian group actions is .
We also give results of similar questions about extendable group actions over
handlebodies.Comment: 22pages, 10 figure
Measurable circle squaring
Laczkovich proved that if bounded subsets and of have the same
non-zero Lebesgue measure and the box dimension of the boundary of each set is
less than , then there is a partition of into finitely many parts that
can be translated to form a partition of . Here we show that it can be
additionally required that each part is both Baire and Lebesgue measurable. As
special cases, this gives measurable and translation-only versions of Tarski's
circle squaring and Hilbert's third problem.Comment: 40 pages; Lemma 4.4 improved & more details added; accepted by Annals
of Mathematic
Automorphism covariant representations of the holonomy-flux *-algebra
We continue an analysis of representations of cylindrical functions and
fluxes which are commonly used as elementary variables of Loop Quantum Gravity.
We consider an arbitrary principal bundle of a compact connected structure
group and following Sahlmann's ideas define a holonomy-flux *-algebra whose
elements correspond to the elementary variables. There exists a natural action
of automorphisms of the bundle on the algebra; the action generalizes the
action of analytic diffeomorphisms and gauge transformations on the algebra
considered in earlier works. We define the automorphism covariance of a
*-representation of the algebra on a Hilbert space and prove that the only
Hilbert space admitting such a representation is a direct sum of spaces L^2
given by a unique measure on the space of generalized connections. This result
is a generalization of our previous work (Class. Quantum. Grav. 20 (2003)
3543-3567, gr-qc/0302059) where we assumed that the principal bundle is
trivial, and its base manifold is R^d.Comment: 34 pages, 1 figure, LaTeX2e, minor clarifying remark
A closure concept in factor-critical graphs
AbstractA graph G is called n-factor-critical if the removal of every set of n vertices results in a~graph with a~1-factor. We prove the following theorem: Let G be a~graph and let x be a~locally n-connected vertex. Let {u,v} be a~pair of vertices in V(G)−{x} such that uv∉E(G), x∈NG(u)∩NG(v), and NG(x)⊂NG(u)∪NG(v)∪{u,v}. Then G is n-factor-critical if and only if G+uv is n-factor-critical
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