800 research outputs found
Composite planar coverings of graphs
AbstractWe shall prove that a connected graph G is projective-planar if and only if it has a 2n-fold planar connected covering obtained as a composition of an n-fold covering and a double covering for some n⩾1 and show that every planar regular covering of a nonplanar graph is such a composite covering
Crossed simplicial groups and structured surfaces
We propose a generalization of the concept of a Ribbon graph suitable to
provide combinatorial models for marked surfaces equipped with a G-structure.
Our main insight is that the necessary combinatorics is neatly captured in the
concept of a crossed simplicial group as introduced, independently, by
Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category
leads to Ribbon graphs while other crossed simplicial groups naturally yield
different notions of structured graphs which model unoriented, N-spin, framed,
etc, surfaces. Our main result is that structured graphs provide orbicell
decompositions of the respective G-structured moduli spaces. As an application,
we show how, building on our theory of 2-Segal spaces, the resulting theory can
be used to construct categorified state sum invariants of G-structured
surfaces.Comment: 86 pages, v2: revised versio
Loops, sign structures and emergent Fermi statistics in three-dimensional quantum dimer models
We introduce and study three-dimensional quantum dimer models with positive
resonance terms. We demonstrate that their ground state wave functions exhibit
a nonlocal sign structure that can be exactly formulated in terms of loops, and
as a direct consequence, monomer excitations obey Fermi statistics. The sign
structure and Fermi statistics in these "signful" quantum dimer models can be
naturally described by a parton construction, which becomes exact at the
solvable point.Comment: 9 pages, 12 figure
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Dimer and fermionic formulations of a class of colouring problems
We show that the number Z of q-edge-colourings of a simple regular graph of
degree q is deducible from functions describing dimers on the same graph, viz.
the dimer generating function or equivalently the set of connected dimer
correlation functions. Using this relationship to the dimer problem, we derive
fermionic representations for Z in terms of Grassmann integrals with quartic
actions. Expressions are given for planar graphs and for nonplanar graphs
embeddable (without edge crossings) on a torus. We discuss exact numerical
evaluations of the Grassmann integrals using an algorithm by Creutz, and
present an application to the 4-edge-colouring problem on toroidal square
lattices, comparing the results to numerical transfer matrix calculations and a
previous Bethe ansatz study. We also show that for the square, honeycomb, 3-12,
and one-dimensional lattice, known exact results for the asymptotic scaling of
Z with the number of vertices can be expressed in a unified way as different
values of one and the same function.Comment: 16 pages, 2 figures, 2 tables. v2: corrected an inconsistency in the
notatio
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