2,008 research outputs found
On the Roman Bondage Number of Graphs on surfaces
A Roman dominating function on a graph is a labeling such that every vertex with label has a neighbor
with label . The Roman domination number, , of is the
minimum of over such functions. The Roman bondage
number is the cardinality of a smallest set of edges whose removal
from results in a graph with Roman domination number not equal to
. In this paper we obtain upper bounds on in terms of
(a) the average degree and maximum degree, and (b) Euler characteristic, girth
and maximum degree. We also show that the Roman bondage number of every graph
which admits a -cell embedding on a surface with non negative Euler
characteristic does not exceed .Comment: 5 page
Explicit isoperimetric constants and phase transitions in the random-cluster model
The random-cluster model is a dependent percolation model that has
applications in the study of Ising and Potts models. In this paper, several new
results are obtained for the random-cluster model on nonamenable graphs with
cluster parameter . Among these, the main ones are the absence of
percolation for the free random-cluster measure at the critical value, and
examples of planar regular graphs with regular dual where \pc^\f (q) > \pu^\w
(q) for large enough. The latter follows from considerations of
isoperimetric constants, and we give the first nontrivial explicit calculations
of such constants. Such considerations are also used to prove non-robust phase
transition for the Potts model on nonamenable regular graphs
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