Helly’s Theorem and its relatives, the theorems of Radon and Caratheodory have played an important role in Discrete and Convex Geometry, there are numerous generalizations and variations of them that have been studied from different perspectives and points of view. The classical lemma of Radon for instance states that any d + 2 points in R can be partitioned into two classes with intersecting c...

We study the relationship between unit-distance representations and the Lovász theta number of graphs, originally established by Lovász. We derive and prove min-max theorems. This framework allows us to derive a weighted version of the hypersphere number of a graph and a related min-max theorem. Then, we connect to sandwich theorems via graph homomorphisms. We present and study a generalization...

A k-weighted graph is a directed graph where each vertex has k associated weights. Partitioning such a graph so that each weight is roughly evenly divided into each piece is useful for balancing a parallel task with k distinct components. Such a goal arises in crash simulation, in which there there are 2 components. We present a 2-weighted graph partitioning algorithm, based on the Ham Sandwich...

A graph G = (V,E) is a chordal probe graph if its vertices can be partitioned into two sets, P (probes) and N (non-probes), where N is a stable set and such that G can be extended to a chordal graph by adding edges between non-probes. We give several characterizations of chordal probe graphs, first, in the case of a fixed given partition of the vertices into probes and non-probes, and second, i...

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product [X × H × Y ]σ of a topological group H over topological spaces X, Y with a continuous sandwich function σ : Y × X → H. We prove that a simple topological semigroup S is a topological paragroup if one of the following conditions is satisfied: (1) S is completely simple and the maximal subgrou...

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product [X × H × Y ]σ of a topological group H over topological spaces X, Y with a continuous sandwich function σ : Y × X → H. We prove that a simple topological semigroup S is a topological paragroup if one of the following conditions is satisfied: (1) S is completely simple and the maximal subgrou...

Abstract A thin Lie algebra is a $L$ , graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating such that each non-zero ideal lies between consecutive terms lower central series. All components have one or two, two-dimensional are called diamonds. Suppose second diamond (that is, next past ) occurs in degree $k$ . We prove if $k>5$ then $[...

The chain graph sandwich problem asks: Given a vertex set V , a mandatory edge set E1, and a larger edge set E2, is there a graph G= (V ,E) such that E1 ⊆ E ⊆ E2 with G being a chain graph (i.e., a 2K2-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E1 is a connected graph, which has a linear-time solutio...

The aim of this work is to evaluate the influence of fabrication methods on the com-pressive properties of sandwich composites using both experimental work and numerical simulations. Two types of sandwich composites with E-glass/Kevlar/polyester resin facings and End-grain balsa wood as core have been produced by vacuum infusion processing (VIP) and hand-lay up (HL) method. Compression tests on...