5,380 research outputs found
On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings
The problem of establishing the number of perfect matchings necessary to
cover the edge-set of a cubic bridgeless graph is strictly related to a famous
conjecture of Berge and Fulkerson. In this paper we prove that deciding whether
this number is at most 4 for a given cubic bridgeless graph is NP-complete. We
also construct an infinite family of snarks (cyclically
4-edge-connected cubic graphs of girth at least five and chromatic index four)
whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs
were known. It turns out that the family also has interesting
properties with respect to the shortest cycle cover problem. The shortest cycle
cover of any cubic bridgeless graph with edges has length at least
, and we show that this inequality is strict for graphs of .
We also construct the first known snark with no cycle cover of length less than
.Comment: 17 pages, 8 figure
A Reciprocity Theorem for Monomer-Dimer Coverings
The problem of counting monomer-dimer coverings of a lattice is a
longstanding problem in statistical mechanics. It has only been exactly solved
for the special case of dimer coverings in two dimensions. In earlier work,
Stanley proved a reciprocity principle governing the number of dimer
coverings of an by rectangular grid (also known as perfect matchings),
where is fixed and is allowed to vary. As reinterpreted by Propp,
Stanley's result concerns the unique way of extending to so
that the resulting bi-infinite sequence, for , satisfies a
linear recurrence relation with constant coefficients. In particular, Stanley
shows that is always an integer satisfying the relation where unless 2(mod 4) and
is odd, in which case . Furthermore, Propp's method is
applicable to higher-dimensional cases. This paper discusses similar
investigations of the numbers , of monomer-dimer coverings, or
equivalently (not necessarily perfect) matchings of an by rectangular
grid. We show that for each fixed there is a unique way of extending
to so that the resulting bi-infinite sequence, for , satisfies a linear recurrence relation with constant coefficients. We
show that , a priori a rational number, is always an integer, using a
generalization of the combinatorial model offered by Propp. Lastly, we give a
new statement of reciprocity in terms of multivariate generating functions from
which Stanley's result follows.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete
Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes
Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems
Friedland's Lower Matching Conjecture asserts that if is a --regular
bipartite graph on vertices, and denotes the number of
matchings of size , then where . When
, this conjecture reduces to a theorem of Schrijver which says that a
--regular bipartite graph on vertices has at least
perfect matchings. L. Gurvits
proved an asymptotic version of the Lower Matching Conjecture, namely he proved
that
In this paper, we prove the Lower Matching Conjecture. In fact, we will prove
a slightly stronger statement which gives an extra factor
compared to the conjecture if is separated away from and , and is
tight up to a constant factor if is separated away from . We will also
give a new proof of Gurvits's and Schrijver's theorems, and we extend these
theorems to --biregular bipartite graphs
- …