15 research outputs found
Hamiltonian-connectedness of triangulations with few separating triangles
We prove that 3-connected plane triangulations containing a single edge contained in all separating triangles are hamiltonian-connected. As a direct corollary we have that 3-connected plane triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any s >= 4 there are 3-connected triangulation with s separating triangles that are not hamiltonian-connected. We also present computational results which show that all `small' 3-connected triangulations with at most 3 separating triangles are hamiltonian-connected
Hamiltonian cycles and 1-factors in 5-regular graphs
It is proven that for any integer and ,
there exist infinitely many 5-regular graphs of genus containing a
1-factorisation with exactly pairs of 1-factors that are perfect, i.e. form
a hamiltonian cycle. For , this settles a problem of Kotzig from 1964.
Motivated by Kotzig and Labelle's "marriage" operation, we discuss two gluing
techniques aimed at producing graphs of high cyclic edge-connectivity. We prove
that there exist infinitely many planar 5-connected 5-regular graphs in which
every 1-factorisation has zero perfect pairs. On the other hand, by the Four
Colour Theorem and a result of Brinkmann and the first author, every planar
4-connected 5-regular graph satisfying a condition on its hamiltonian cycles
has a linear number of 1-factorisations each containing at least one perfect
pair. We also prove that every planar 5-connected 5-regular graph satisfying a
stronger condition contains a 1-factorisation with at most nine perfect pairs,
whence, every such graph admitting a 1-factorisation with ten perfect pairs has
at least two edge-Kempe equivalence classes. The paper concludes with further
results on edge-Kempe equivalence classes in planar 5-regular graphs.Comment: 27 pages, 13 figures; corrected figure
On the genera of polyhedral embeddings of cubic graph
In this article we present theoretical and computational results on the
existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs.
We also describe an efficient algorithm to compute all polyhedral embeddings of
a given cubic graph and constructions for cubic graphs with some special
properties of their polyhedral embeddings. Some key results are that even cubic
graphs with a polyhedral embedding on the torus can also have polyhedral
embeddings in arbitrarily high genus, in fact in a genus {\em close} to the
theoretical maximum for that number of vertices, and that there is no bound on
the number of genera in which a cubic graph can have a polyhedral embedding.
While these results suggest a large variety of polyhedral embeddings,
computations for up to 28 vertices suggest that by far most of the cubic graphs
do not have a polyhedral embedding in any genus and that the ratio of these
graphs is increasing with the number of vertices.Comment: The C-program implementing the algorithm described in this article
can be obtained from any of the author
10-Gabriel graphs are Hamiltonian
Given a set of points in the plane, the -Gabriel graph of is the
geometric graph with vertex set , where are connected by an
edge if and only if the closed disk having segment as diameter
contains at most points of . We consider the
following question: What is the minimum value of such that the -Gabriel
graph of every point set contains a Hamiltonian cycle? For this value, we
give an upper bound of 10 and a lower bound of 2. The best previously known
values were 15 and 1, respectively
Forcing Independence
An independent set in a graph is a set of vertices which are pairwise non-adjacent. An independ-ent set of vertices F is a forcing independent set if there is a unique maximum independent set I such that F ⊆ I. The forcing independence number or forcing number of a maximum independent set I is the cardi-nality of a minimum forcing set for I. The forcing number f of a graph is the minimum cardinality of the forcing numbers for the maximum independent sets of the graph. The possible values of f are determined and characterized. We investigate connections between these concepts, other structural concepts, and chemical applications. (doi: 10.5562/cca2295
On the minimum leaf number of cubic graphs
The \emph{minimum leaf number} of a connected graph is
defined as the minimum number of leaves of the spanning trees of . We
present new results concerning the minimum leaf number of cubic graphs: we show
that if is a connected cubic graph of order , then , improving on the best known result in [Inf. Process.
Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph
Theory and Applications 5 (2017) 207-211]. We further prove that if is also
2-connected, then , improving on the best
known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new
conjectures concerning the minimum leaf number of several types of cubic graphs
and examples showing that the bounds of the conjectures are best possible.Comment: 17 page