It is shown that every planar graph with no separating triangles is a subgraph of a Hamiltonian planar graph; that is, Whitney’s theorem holds without the assumption of a triangulation.

We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers‘ triangular array are then calculated and the explicit formula for them is provided. Next in the second construction we endow the set sums of pr...

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d . We derive a general formula for the representation of the symmetric group on the homology of posets of pa...

The idea of enumeration using the Euler characteristic was suggested throughout Rota’s work and influenced by Schanuel’s categorical viewpoint [33, 40, 41, 42]. In order to carry out such a program that is topologically meaningful and which captures the broadest possible classes of examples, two key insights are required. First, the notion of grading in the face lattice of a polytope must be re...

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney...

This paper investigates the application of the Whitney Reduction Network (WRN) to the low-dimensional characterization of digital images human faces. Motivated by Whitney's Embedding theorem from diierential topology the WRN provides a non-linear parameterization of m dimensional manifolds. Based on this, the reduction of the high-dimensional raw image data consists of two-stages. First, a low-...

The concept of matroid, with its companion concept of geometric lattice, was distilled by Hassler Whitney [19], Saunders Mac Lane [10] and Garrett Birkhoff [2] from the common properties of linear and algebraic dependence. The inverse problem, how to represent a given abstract matroid as the matroid of linear dependence of a specified set of vectors over some field (or as the matroid of algebra...

Let {X1, . . . , Xm} and {Y1, . . . , Yn} be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functions F and G, respectively. We study the properties of the classical WilcoxonMann-Whitney statistic for testing for stochastic dominance in the above set up.