222 research outputs found
Types of triangle in plane Hamiltonian triangulations and applications to domination and k-walks
We investigate the minimum number t(0)(G) of faces in a Hamiltonian triangulation G so that any Hamiltonian cycle C of G has at least t(0)(G) faces that do not contain an edge of C. We prove upper and lower bounds on the maximum of these numbers for all triangulations with a fixed number of facial triangles. Such triangles play an important role when Hamiltonian cycles in triangulations with 3-cuts are constructed from smaller Hamiltonian cycles of 4-connected subgraphs. We also present results linking the number of these triangles to the length of 3-walks in a class of triangulation and to the domination number
A History of Flips in Combinatorial Triangulations
Given two combinatorial triangulations, how many edge flips are necessary and
sufficient to convert one into the other? This question has occupied
researchers for over 75 years. We provide a comprehensive survey, including
full proofs, of the various attempts to answer it.Comment: Added a paragraph referencing earlier work in the vertex-labelled
setting that has implications for the unlabeled settin
Polyhedra with few 3-cuts are hamiltonian
In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In
this article, we will generalize this result and prove that polyhedra with at
most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this
result for the subclass of triangulations. We also prove that polyhedra with at
most four 3-cuts have a hamiltonian path. It is well known that for each non-hamiltonian polyhedra with 3-cuts exist. We give computational
results on lower bounds on the order of a possible non-hamiltonian polyhedron
for the remaining open cases of polyhedra with four or five 3-cuts.Comment: 21 pages; changed titl
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
Thoughts on Barnette's Conjecture
We prove a new sufficient condition for a cubic 3-connected planar graph to
be Hamiltonian. This condition is most easily described as a property of the
dual graph. Let be a planar triangulation. Then the dual is a cubic
3-connected planar graph, and is bipartite if and only if is
Eulerian. We prove that if the vertices of are (improperly) coloured blue
and red, such that the blue vertices cover the faces of , there is no blue
cycle, and every red cycle contains a vertex of degree at most 4, then is
Hamiltonian.
This result implies the following special case of Barnette's Conjecture: if
is an Eulerian planar triangulation, whose vertices are properly coloured
blue, red and green, such that every red-green cycle contains a vertex of
degree 4, then is Hamiltonian. Our final result highlights the
limitations of using a proper colouring of as a starting point for proving
Barnette's Conjecture. We also explain related results on Barnette's Conjecture
that were obtained by Kelmans and for which detailed self-contained proofs have
not been published.Comment: 12 pages, 7 figure
Counting cycles in planar triangulations
We investigate the minimum number of cycles of specified lengths in planar
-vertex triangulations . It is proven that this number is for
any cycle length at most , where denotes the radius
of the triangulation's dual, which is at least logarithmic but can be linear in
the order of the triangulation. We also show that there exist planar
hamiltonian -vertex triangulations containing many -cycles for any
. Furthermore, we prove
that planar 4-connected -vertex triangulations contain many
-cycles for every , and that, under certain
additional conditions, they contain -cycles for many values of
, including
Polyhedra with few 3-cuts are hamiltonian
In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In this article, we will generalize this result and prove that polyhedra with at most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four 3-cuts have a hamiltonian path. It is well known that for each k ≥ 6 non-hamiltonian polyhedra with k 3-cuts exist. We give computational results on lower bounds on the order of a possible non-hamiltonian polyhedron for the remaining open cases of polyhedra with four or five 3-cuts
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
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