564 research outputs found
A counterexample to the pseudo 2-factor isomorphic graph conjecture
A graph is pseudo 2-factor isomorphic if the parity of the number of
cycles in a 2-factor is the same for all 2-factors of . Abreu et al.
conjectured that , the Heawood graph and the Pappus graph are the only
essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs
(Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture
3.6).
Using a computer search we show that this conjecture is false by constructing
a counterexample with 30 vertices. We also show that this is the only
counterexample up to at least 40 vertices.
A graph is 2-factor hamiltonian if all 2-factors of are hamiltonian
cycles. Funk et al. conjectured that every 2-factor hamiltonian cubic bipartite
graph can be obtained from and the Heawood graph by applying repeated
star products (Funk et al., Journal of Combinatorial Theory, Series B, 2003,
Conjecture 3.2). We verify that this conjecture holds up to at least 40
vertices.Comment: 8 pages, added some extra information in Discrete Applied Mathematics
(2015
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
More on quasi-random graphs, subgraph counts and graph limits
We study some properties of graphs (or, rather, graph sequences) defined by
demanding that the number of subgraphs of a given type, with vertices in
subsets of given sizes, approximatively equals the number expected in a random
graph. It has been shown by several authors that several such conditions are
quasi-random, but that there are exceptions. In order to understand this
better, we investigate some new properties of this type. We show that these
properties too are quasi-random, at least in some cases; however, there are
also cases that are left as open problems, and we discuss why the proofs fail
in these cases.
The proofs are based on the theory of graph limits; and on the method and
results developed by Janson (2011), this translates the combinatorial problem
to an analytic problem, which then is translated to an algebraic problem.Comment: 35 page
- …