18,334 research outputs found
On Almost Well-Covered Graphs of Girth at Least 6
We consider a relaxation of the concept of well-covered graphs, which are
graphs with all maximal independent sets of the same size. The extent to which
a graph fails to be well-covered can be measured by its independence gap,
defined as the difference between the maximum and minimum sizes of a maximal
independent set in . While the well-covered graphs are exactly the graphs of
independence gap zero, we investigate in this paper graphs of independence gap
one, which we also call almost well-covered graphs. Previous works due to
Finbow et al. (1994) and Barbosa et al. (2013) have implications for the
structure of almost well-covered graphs of girth at least for . We focus on almost well-covered graphs of girth at least . We show
that every graph in this class has at most two vertices each of which is
adjacent to exactly leaves. We give efficiently testable characterizations
of almost well-covered graphs of girth at least having exactly one or
exactly two such vertices. Building on these results, we develop a
polynomial-time recognition algorithm of almost well-covered
-free graphs
On Cyclic Edge-Connectivity of Fullerenes
A graph is said to be cyclic -edge-connected, if at least edges must
be removed to disconnect it into two components, each containing a cycle. Such
a set of edges is called a cyclic--edge cutset and it is called a
trivial cyclic--edge cutset if at least one of the resulting two components
induces a single -cycle.
It is known that fullerenes, that is, 3-connected cubic planar graphs all of
whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this
article it is shown that a fullerene containing a nontrivial cyclic-5-edge
cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces
whose neighboring faces are also pentagonal. Moreover, it is shown that has
a Hamilton cycle, and as a consequence at least perfect matchings, where is the order of .Comment: 11 pages, 9 figure
Bootstrap percolation on the Hamming torus
The Hamming torus of dimension is the graph with vertices
and an edge between any two vertices that differ in a single
coordinate. Bootstrap percolation with threshold starts with a random
set of open vertices, to which every vertex belongs independently with
probability , and at each time step the open set grows by adjoining every
vertex with at least open neighbors. We assume that is large and
that scales as for some , and study the probability
that an -dimensional subgraph ever becomes open. For large , we
prove that the critical exponent is about for , and
about for . Our small
results are mostly limited to , where we identify the critical in
many cases and, when , compute exactly the critical probability that
the entire graph is eventually open.Comment: Published in at http://dx.doi.org/10.1214/13-AAP996 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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