14,798 research outputs found
Large Non-Planar Graphs and an Application to Crossing-Critical Graphs
We prove that, for every positive integer k, there is an integer N such that
every 4-connected non-planar graph with at least N vertices has a minor
isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding
an edge joining every pair of vertices at distance exactly k, or the graph
obtained from a cycle of length k by adding two vertices adjacent to each other
and to every vertex on the cycle. We also prove a version of this for
subdivisions rather than minors, and relax the connectivity to allow 3-cuts
with one side planar and of bounded size. We deduce that for every integer k
there are only finitely many 3-connected 2-crossing-critical graphs with no
subdivision isomorphic to the graph obtained from a cycle of length 2k by
joining all pairs of diagonally opposite vertices.Comment: To appear in Journal of Combinatorial Theory B. 20 pages. No figures.
Te
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
Planar Ising model at criticality: state-of-the-art and perspectives
In this essay, we briefly discuss recent developments, started a decade ago
in the seminal work of Smirnov and continued by a number of authors, centered
around the conformal invariance of the critical planar Ising model on
and, more generally, of the critical Z-invariant Ising model on
isoradial graphs (rhombic lattices). We also introduce a new class of
embeddings of general weighted planar graphs (s-embeddings), which might, in
particular, pave the way to true universality results for the planar Ising
model.Comment: 19 pages (+ references), prepared for the Proceedings of ICM2018.
Second version: two references added, a few misprints fixe
Rapid algorithm for identifying backbones in the two-dimensional percolation model
We present a rapid algorithm for identifying the current-carrying backbone in
the percolation model. It applies to general two-dimensional graphs with open
boundary conditions. Complemented by the modified Hoshen-Kopelman cluster
labeling algorithm, our algorithm identifies dangling parts using their local
properties. For planar graphs, it finds the backbone almost four times as fast
as Tarjan's depth-first-search algorithm, and uses the memory of the same size
as the modified Hoshen-Kopelman algorithm. Comparison with other algorithms for
backbone identification is addressed.Comment: 5 pages with 5 eps figures. RevTeX 3.1. Clarify the origin of the
hull-generating algorith
- …