30 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
A note on 5-cycle double covers
The strong cycle double cover conjecture states that for every circuit of
a bridgeless cubic graph , there is a cycle double cover of which
contains . We conjecture that there is even a 5-cycle double cover of
which contains , i.e. is a subgraph of one of the five 2-regular
subgraphs of . We prove a necessary and sufficient condition for a 2-regular
subgraph to be contained in a 5-cycle double cover of
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
Multiple Petersen subdivisions in permutation graphs
A permutation graph is a cubic graph admitting a 1-factor M whose complement
consists of two chordless cycles. Extending results of Ellingham and of
Goldwasser and Zhang, we prove that if e is an edge of M such that every
4-cycle containing an edge of M contains e, then e is contained in a
subdivision of the Petersen graph of a special type. In particular, if the
graph is cyclically 5-edge-connected, then every edge of M is contained in such
a subdivision. Our proof is based on a characterization of cographs in terms of
twin vertices. We infer a linear lower bound on the number of Petersen
subdivisions in a permutation graph with no 4-cycles, and give a construction
showing that this lower bound is tight up to a constant factor
Covering cubic graphs with matchings of large size
Let m be a positive integer and let G be a cubic graph of order 2n. We
consider the problem of covering the edge-set of G with the minimum number of
matchings of size m. This number is called excessive [m]-index of G in
literature. The case m=n, that is a covering with perfect matchings, is known
to be strictly related to an outstanding conjecture of Berge and Fulkerson. In
this paper we study in some details the case m=n-1. We show how this parameter
can be large for cubic graphs with low connectivity and we furnish some
evidence that each cyclically 4-connected cubic graph of order 2n has excessive
[n-1]-index at most 4. Finally, we discuss the relation between excessive
[n-1]-index and some other graph parameters as oddness and circumference.Comment: 11 pages, 5 figure