3,186 research outputs found
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
Cyclically five-connected cubic graphs
A cubic graph is cyclically 5-connected if is simple, 3-connected,
has at least 10 vertices and for every set of edges of size at most four,
at most one component of contains circuits. We prove that if
and are cyclically 5-connected cubic graphs and topologically
contains , then either and are isomorphic, or (modulo well-described
exceptions) there exists a cyclically 5-connected cubic graph such that
topologically contains and is obtained from in one of the
following two ways. Either is obtained from by subdividing two
distinct edges of and joining the two new vertices by an edge, or is
obtained from by subdividing each edge of a circuit of length five and
joining the new vertices by a matching to a new circuit of length five disjoint
from in such a way that the cyclic orders of the two circuits agree. We
prove a companion result, where by slightly increasing the connectivity of
we are able to eliminate the second construction. We also prove versions of
both of these results when is almost cyclically 5-connected in the sense
that it satisfies the definition except for 4-edge cuts such that one side is a
circuit of length four. In this case is required to be almost cyclically
5-connected and to have fewer circuits of length four than . In particular,
if has at most one circuit of length four, then is required to be
cyclically 5-connected. However, in this more general setting the operations
describing the possible graphs are more complicated.Comment: 47 pages, 5 figures. Revised according to referee's comments. To
appear in J. Combin. Theory Ser.
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
Generation and Properties of Snarks
For many of the unsolved problems concerning cycles and matchings in graphs
it is known that it is sufficient to prove them for \emph{snarks}, the class of
nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part
of this paper we present a new algorithm for generating all non-isomorphic
snarks of a given order. Our implementation of the new algorithm is 14 times
faster than previous programs for generating snarks, and 29 times faster for
generating weak snarks. Using this program we have generated all non-isomorphic
snarks on vertices. Previously lists up to vertices have been
published. In the second part of the paper we analyze the sets of generated
snarks with respect to a number of properties and conjectures. We find that
some of the strongest versions of the cycle double cover conjecture hold for
all snarks of these orders, as does Jaeger's Petersen colouring conjecture,
which in turn implies that Fulkerson's conjecture has no small counterexamples.
In contrast to these positive results we also find counterexamples to eight
previously published conjectures concerning cycle coverings and the general
cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated
and typos corrected. This version differs from the published one in that the
Arxiv-version has data about the automorphisms of snarks; Journal of
Combinatorial Theory. Series B. 201
Poorly connected groups
We investigate groups whose Cayley graphs have poor\-ly connected subgraphs.
We prove that a finitely generated group has bounded separation in the sense of
Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove
a gap theorem for connectivity of finitely presented groups, and prove that
there is no comparable theorem for all finitely generated groups. Finally, we
formulate a connectivity version of the conjecture that every group of type
with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with
at most quadratic Dehn function.Comment: 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem
1.4 added completing the no-gap result outlined in v
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