A Remark on Partitions and Triangles

On Tensor Product of Graphs, Girth and Triangles

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

On Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their c...

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-triangles

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their c...

A Remark on Prime Divisors of Lengths of Sides of Heron Triangles

A Heron triangle is a triangle having the property that the lengths of its sides as well as its area are positive integers. Let P be a fixed set of primes, let S denote the set of integers divisible only by primes in P. We prove that there are only finitely many Heron triangles whose sides a, b, c ∈ S and are reduced, that is gcd(a, b, c) = 1. If P contains only one prime ≡ 1 (mod 4) then all t...