125 research outputs found
Classification of finite groups with toroidal or projective-planar permutability graphs
Let be a group. The permutability graph of subgroups of , denoted by
, is a graph having all the proper subgroups of as its vertices,
and two subgroups are adjacent in if and only if they permute. In
this paper, we classify the finite groups whose permutability graphs are
toroidal or projective-planar. In addition, we classify the finite groups whose
permutability graph does not contain one of , , , ,
or as a subgraph.Comment: 30 pages, 8 figure
The PC-Tree algorithm, Kuratowski subdivisions, and the torus.
The PC-Tree algorithm of Shih and Hsu (1999) is a practical linear-time planarity algorithm that provides a plane embedding of the given graph if it is planar and a Kuratowski subdivision otherwise. Remarkably, there is no known linear-time algorithm for embedding graphs on the torus. We extend the PC-Tree algorithm to a practical, linear-time toroidality test for K3;3-free graphs called the PCK-Tree algorithm. We also prove that it is NP-complete to decide whether the edges of a graph can be covered with two Kuratowski subdivisions. This greatly reduces the possibility of a polynomial-time toroidality testing algorithm based solely on edge-coverings by subdivisions of Kuratowski subgraphs
Minor-Obstructions for Apex-Pseudoforests
A graph is called a pseudoforest if none of its connected components contains
more than one cycle. A graph is an apex-pseudoforest if it can become a
pseudoforest by removing one of its vertices. We identify 33 graphs that form
the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of
all minor-minimal graphs that are not apex-pseudoforests
Tight factorizations of girth--regular graphs
The determination of 1-factorizations of girth-regular graphs of girth,
regular degree and chromatic index is proposed for the cases in which each
girth cycle intersects every 1-factor of . This endeavor may apply to
priority assignment problems, managerial situations in optimization and
decision making. Applications to hamiltonian decomposability (via union of
pairs of 1-factors) and to 3-dimensional geometry (M\"obius-strip compounds and
hollow-triangle polylinks) are given.Comment: 37 pages, 19 figures, 10 tables05C
Holomorphic Anomalies in Topological Field Theories
We study the stringy genus one partition function of SCFT's. It is
shown how to compute this using an anomaly in decoupling of BRST trivial states
from the partition function. A particular limit of this partition function
yields the partition function of topological theory coupled to topological
gravity. As an application we compute the number of holomorphic elliptic curves
over certain Calabi-Yau manifolds including the quintic threefold. This may be
viewed as the first application of mirror symmetry at the string quantum level.Comment: 32 pages. Appendix by S.Kat
- …