8,898 research outputs found
Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
We settle a problem of Havel by showing that there exists an absolute
constant d such that if G is a planar graph in which every two distinct
triangles are at distance at least d, then G is 3-colorable. In fact, we prove
a more general theorem. Let G be a planar graph, and let H be a set of
connected subgraphs of G, each of bounded size, such that every two distinct
members of H are at least a specified distance apart and all triangles of G are
contained in \bigcup{H}. We give a sufficient condition for the existence of a
3-coloring phi of G such that for every B\in H, the restriction of phi to B is
constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio
Rapid mixing of Swendsen-Wang dynamics in two dimensions
We prove comparison results for the Swendsen-Wang (SW) dynamics, the
heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics
for the random-cluster model on arbitrary graphs. In particular, we prove that
rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum
degree and that rapid mixing of SW and rapid mixing of SB are equivalent.
Additionally, the spectral gap of SW and SB on planar graphs is bounded from
above and from below by the spectral gap of these dynamics on the corresponding
dual graph with suitably changed temperature.
As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the
Potts model on the two-dimensional square lattice at all non-critical
temperatures as well as rapid mixing for the two-dimensional Ising model at all
temperatures. Furthermore, we obtain new results for general graphs at high or
low enough temperatures.Comment: Ph.D. thesis, 66 page
Characterizing 2-crossing-critical graphs
It is very well-known that there are precisely two minimal non-planar graphs:
and (degree 2 vertices being irrelevant in this context). In
the language of crossing numbers, these are the only 1-crossing-critical
graphs: they each have crossing number at least one, and every proper subgraph
has crossing number less than one. In 1987, Kochol exhibited an infinite family
of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i)
determine all the 3-connected 2-crossing-critical graphs that contain a
subdivision of the M\"obius Ladder ; (ii) show how to obtain all the
not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii)
show that there are only finitely many 3-connected 2-crossing-critical graphs
not containing a subdivision of ; and (iv) determine all the
3-connected 2-crossing-critical graphs that do not contain a subdivision of
.Comment: 176 pages, 28 figure
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