14 research outputs found
Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs
We construct several families of genus embeddings of near-complete graphs
using index 2 current graphs. In particular, we give the first known minimum
genus embeddings of certain families of octahedral graphs, solving a
longstanding conjecture of Jungerman and Ringel, and Hamiltonian cycle
complements, making partial progress on a problem of White. Index 2 current
graphs are also applied to various cases of the Map Color Theorem, in some
cases yielding significantly simpler solutions, e.g., the nonorientable genus
of . We give a complete description of the method, originally
due to Jungerman, from which all these results were obtained.Comment: 23 pages, 21 figures; fixed 2 figures from previous versio
The Graph Isomorphism Problem and approximate categories
It is unknown whether two graphs can be tested for isomorphism in polynomial
time. A classical approach to the Graph Isomorphism Problem is the
d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can
distinguish many pairs of graphs, but the pairs of non-isomorphic graphs
constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed,
then the WL-algorithm runs in polynomial time. We will formulate the Graph
Isomorphism Problem as an Orbit Problem: Given a representation V of an
algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie
in the same G-orbit. Then we attack the Orbit Problem by constructing certain
approximate categories C_d(V), d=1,2,3,... whose objects include the elements
of V. We show that v_1 and v_2 are not in the same orbit by showing that they
are not isomorphic in the category C_d(V) for some d. For every d this gives us
an algorithm for isomorphism testing. We will show that the WL-algorithms
reduce to our algorithms, but that our algorithms cannot be reduced to the
WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can
distinguish the Cai-Furer-Immerman graphs in polynomial time.Comment: 29 page
Recommended from our members
Testing Convexity and Acyclicity, and New Constructions for Dense Graph Embeddings
Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree.
Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces
Real equiangular lines and related codes
We consider real equiangular lines and related codes. The driving question is to find the maximum number of equiangular lines in a given dimension. In the real case, this is controlled by combinatorial phenomena, and until only very recently, the exact number has been unknown. The complex case appears to be driven by other phenomena, and configurations are conjectured always to meet the absolute bound of d^2 lines in dimension d. We consider a variety of the techniques that have been used to approach the problem, both for constructing large sets of equiangular lines, and for finding tighter upper bounds. Many of the best-known upper bounds for codes are instances of a general linear programming bound, which we discuss in detail. At various points throughout the thesis, we note applications in quantum information theory
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4