209 research outputs found
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
Cubic graphs with large circumference deficit
The circumference of a graph is the length of a longest cycle. By
exploiting our recent results on resistance of snarks, we construct infinite
classes of cyclically -, - and -edge-connected cubic graphs with
circumference ratio bounded from above by , and
, respectively. In contrast, the dominating cycle conjecture implies
that the circumference ratio of a cyclically -edge-connected cubic graph is
at least .
In addition, we construct snarks with large girth and large circumference
deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On
stable cycles and cycle double covers of graphs with large circumference, Disc.
Math. 312 (2012), 2540--2544]
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