20,692 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
Critical classes, Kronecker products of spin characters, and the Saxl conjecture
Highlighting the use of critical classes, we consider constituents in
Kronecker products, in particular of spin characters of the double covers of
the symmetric and alternating groups. We apply results from the spin case to
find constituents in Kronecker products of characters of the symmetric groups.
Via this tool, we make progress on the Saxl conjecture; this claims that for a
triangular number , the square of the irreducible character of the symmetric
group labelled by the staircase contains all irreducible characters of
as constituents. We find a large number of constituents in this square
which were not detected by other methods. Moreover, the investigation of
Kronecker products of spin characters inspires a spin variant of Saxl's
conjecture.Comment: 17 page
Construction of cycle double covers for certain classes of graphs
We introduce two classes of graphs, Indonesian graphs and -doughnut graphs. Cycle double covers are constructed for these classes. In case of doughnut graphs this is done for the values and 4
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