117 research outputs found
Some snarks are worse than others
Many conjectures and open problems in graph theory can either be reduced to
cubic graphs or are directly stated for cubic graphs. Furthermore, it is known
that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless
cubic graph which is not 3--edge-colourable. In this paper we deal with the
fact that the family of potential counterexamples to many interesting
conjectures can be narrowed even further to the family of
bridgeless cubic graphs whose edge set cannot be covered with four perfect
matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover
Conjecture and the Fan-Raspaud Conjecture are examples of statements for which
is crucial. In this paper, we study parameters which have
the potential to further refine and thus enlarge the set of
cubic graphs for which the mentioned conjectures can be verified. We show that
can be naturally decomposed into subsets with increasing
complexity, thereby producing a natural scale for proving these conjectures.
More precisely, we consider the following parameters and questions: given a
bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii)
how many copies of the same perfect matching need to be added, and (iii) how
many 2--factors need to be added so that the resulting regular graph is Class
I? We present new results for these parameters and we also establish some
strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure
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
Covering a cubic graph with perfect matchings
Let G be a bridgeless cubic graph. A well-known conjecture of Berge and
Fulkerson can be stated as follows: there exist five perfect matchings of G
such that each edge of G is contained in at least one of them. Here, we prove
that in each bridgeless cubic graph there exist five perfect matchings covering
a portion of the edges at least equal to 215/231 . By a generalization of this
result, we decrease the best known upper bound, expressed in terms of the size
of the graph, for the number of perfect matchings needed to cover the edge-set
of G.Comment: accepted for the publication in Discrete Mathematic
Normal 6-edge-colorings of some bridgeless cubic graphs
In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of
colors assigned to the edge and the four edges adjacent it, has exactly five or
exactly three distinct colors, respectively. An edge is normal in an
edge-coloring if it is rich or poor in this coloring. A normal
-edge-coloring of a cubic graph is an edge-coloring with colors such
that each edge of the graph is normal. We denote by the smallest
, for which admits a normal -edge-coloring. Normal edge-colorings
were introduced by Jaeger in order to study his well-known Petersen Coloring
Conjecture. It is known that proving for every bridgeless
cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover,
Jaeger was able to show that it implies classical conjectures like Cycle Double
Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors
were able to show that any simple cubic graph admits a normal
-edge-coloring, and this result is best possible. In the present paper, we
show that any claw-free bridgeless cubic graph, permutation snark, tree-like
snark admits a normal -edge-coloring. Finally, we show that any bridgeless
cubic graph admits a -edge-coloring such that at least edges of are normal.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1804.0944
Normal edge-colorings of cubic graphs
A normal -edge-coloring of a cubic graph is an edge-coloring with
colors having the additional property that when looking at the set of colors
assigned to any edge and the four edges adjacent it, we have either exactly
five distinct colors or exactly three distinct colors. We denote by
the smallest , for which admits a normal
-edge-coloring. Normal -edge-colorings were introduced by Jaeger in order
to study his well-known Petersen Coloring Conjecture. More precisely, it is
known that proving for every bridgeless cubic graph is
equivalent to proving Petersen Coloring Conjecture and then, among others,
Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the
larger class of all simple cubic graphs (not necessarily bridgeless), some
interesting questions naturally arise. For instance, there exist simple cubic
graphs, not bridgeless, with . On the other hand, the known
best general upper bound for was . Here, we improve it by
proving that for any simple cubic graph , which is best
possible. We obtain this result by proving the existence of specific no-where
zero -flows in -edge-connected graphs.Comment: 17 pages, 6 figure
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
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