20 research outputs found
Construction of planar triangulations with minimum degree 5
AbstractIn this article, we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3-connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum degree 5
Construction of planar 4-connected triangulations
In this article we describe a recursive structure for the class of 4-connected triangulations or - equivalently - cyclically 4-connected plane cubic graphs
On Cyclic Edge-Connectivity of Fullerenes
A graph is said to be cyclic -edge-connected, if at least edges must
be removed to disconnect it into two components, each containing a cycle. Such
a set of edges is called a cyclic--edge cutset and it is called a
trivial cyclic--edge cutset if at least one of the resulting two components
induces a single -cycle.
It is known that fullerenes, that is, 3-connected cubic planar graphs all of
whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this
article it is shown that a fullerene containing a nontrivial cyclic-5-edge
cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces
whose neighboring faces are also pentagonal. Moreover, it is shown that has
a Hamilton cycle, and as a consequence at least perfect matchings, where is the order of .Comment: 11 pages, 9 figure
Wiener Index and Remoteness in Triangulations and Quadrangulations
Let be a a connected graph. The Wiener index of a connected graph is the
sum of the distances between all unordered pairs of vertices. We provide
asymptotic formulae for the maximum Wiener index of simple triangulations and
quadrangulations with given connectivity, as the order increases, and make
conjectures for the extremal triangulations and quadrangulations based on
computational evidence. If denotes the arithmetic mean
of the distances from to all other vertices of , then the remoteness of
is defined as the largest value of over all vertices
of . We give sharp upper bounds on the remoteness of simple
triangulations and quadrangulations of given order and connectivity
Recursive generation of simple planar 5-regular graphs and pentangulations
We describe how the 5-regular simple planar graphs can all be obtained from an elementary family of starting graphs by repeatedly applying a few local expansion operations. The proof uses an amalgam of theory and computation. By incorporating the recursion into the canonical construc- tion path method of isomorph rejection, a generator of non-isomorphic embedded 5-regular planar graphs is obtained with time complexity O(n2) per isomorphism class. A similar result is obtained for simple planar pen- tangulations with minimum degree 2