On the Subsets Product in Finite Groups

Let B be a proper subset of a finite group G such that either B = B−1 or G is abelian. We prove that there exists a subgroup H generated by an element of B with the following property. For every subset A of G such that A ∩ H ≠ ∅, either H ⊂ A ∪ AB or ❘A ∪ AB❘ , ❘A❘ + ❘B❘. This result generalizes the Cauchy-Davenport Theorem and two theorems of Chowla and Shepherdson

Branch-width and well-quasi-ordering in matroids and graphs

AbstractWe prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off

Curve Reconstruction, the Traveling Salesman Problem, and Menger's Theorem on Length

We give necessary and sufficient regularity conditions under which the curve reconstruction problem is solved by a traveling salesman tour or path, respectively. For the proof we have to generalize a theorem of Menger [12], [13] on arc lengt

A PVS Graph Theory Library

This paper documents the NASA Langley PVS graph theory library. The library provides fundamental definitions for graphs, subgraphs, walks, paths, subgraphs generated by walks, trees, cycles, degree, separating sets, and four notions of connectedness. Theorems provided include Ramsey's and Menger's and the equivalence of all four notions of connectedness

Menger Path Systems

https://digitalcommons.memphis.edu/speccoll-faudreerj/1235/thumbnail.jp

Menger\u27s Theorem and Short Paths

https://digitalcommons.memphis.edu/speccoll-faudreerj/1228/thumbnail.jp