1,388 research outputs found
On hypohamiltonian snarks and a theorem of Fiorini
In 2003, Cavicchioli et al. corrected an omission in the statement and proof of Fiorini's theorem from 1983 on hypohamiltonian snarks. However, their version of this theorem contains an unattainable condition for certain cases. We discuss and extend the results of Fiorini and Cavicchioli et al. and present a version of this theorem which is more general in several ways. Using Fiorini's erroneous result, Steffen had shown that hypohamiltonian snarks exist for some orders n >= 10 and each even n >= 92. We rectify Steffen's proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffen's theorem to the strongest possible form by determining all orders for which hypohamiltonian snarks exist. This also strengthens a result of Macajova and Skoviera. Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices
On the smallest snarks with oddness 4 and connectivity 2
A snark is a bridgeless cubic graph which is not 3-edge-colourable. The
oddness of a bridgeless cubic graph is the minimum number of odd components in
any 2-factor of the graph.
Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in [Electron. J.
Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and
remarked that there are exactly two such graphs of that order. However, this
remark is incorrect as -- using an exhaustive computer search -- we show that
there are in fact three snarks with oddness 4 on 28 vertices. In this note we
present the missing snark and also determine all snarks with oddness 4 up to 34
vertices.Comment: 5 page
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
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
Generation of cubic graphs and snarks with large girth
We describe two new algorithms for the generation of all non-isomorphic cubic
graphs with girth at least which are very efficient for
and show how these algorithms can be efficiently restricted to generate snarks
with girth at least .
Our implementation of these algorithms is more than 30, respectively 40 times
faster than the previously fastest generator for cubic graphs with girth at
least 6 and 7, respectively.
Using these generators we have also generated all non-isomorphic snarks with
girth at least 6 up to 38 vertices and show that there are no snarks with girth
at least 7 up to 42 vertices. We present and analyse the new list of snarks
with girth 6.Comment: 27 pages (including appendix
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