Triangle colorings require at least seven colors

We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven colors. This is also true for a coloring using uniformly colored polygons if it has a point bordering at least four polygons.     

Strengthened clique-family inequalities for the stable set polytope

The stable set polytope is a fundamental object in combinatorial optimization. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pêcher and Wagler showed that the clique-family inequalities can be strengthened under certain conditions. We show that they can be strengthened even further, using a surprisingly simple mixed-integer rounding argument.     

Parabolic Kazhdan–Lusztig polynomials for quasi-minuscule quotients

We study the parabolic Kazhdan–Lusztig polynomials for the quasi-minuscule quotients of Weyl groups. We give explicit closed combinatorial formulas for the parabolic Kazhdan–Lusztig polynomials of type q. Our study implies that these are always either zero or a monic power of q, and that they are not combinatorial invariants. We conjecture a combinatorial interpretation for the parabolic Kazhdan–Lusztig polynomials of type −1.     

Skip letters for short supersequence of all permutations

A supersequence over a finite set is a sequence that contains as subsequence all permutations of the set. This paper defines an infinite array of methods to create supersequences of decreasing lengths. This yields the shortest known supersequences over larger sets. It also provides the best results asymptotically. It is based on a general proof using a new property called strong completeness. The same technique also can be used to prove existing supersequences which combines the old and new ones into an unified conceptual framework.     

Preparation and Flame Retardancy of Polyurethane/POSS Nanocomposites

Polyurethane/polyhedral oligomeric sisesquioxane (PU/POSS) nanocomposites were syn-thesized via polymerization utilizing the compatibility between POSS nanoparticles and 4,4'-diphenyl methylene diisocyanate. Scanning electron microscope images and Fouriertransform infrared spectra revealed that POSS nanoparticles were dispersed in PU matrix.Thermal gravimetric analysis was employed to investigate the thermal decomposition be-havior of PU/POSS nanocomposites at elevated temperatures. Then fire performance wasevaluated by limiting oxygen index, underwriters laboratories 94 testing and char residue morphology. These results showed that the addition of POSS promoted the formation of char residues which were covered on the surface of polymer composites, leading to the im-provement of thermal stability and flame retardancy.     

The unambiguity of segmented morphisms

This paper studies the ambiguity of morphisms in free monoids. A morphism σ is said to be ambiguous with respect to a string α if there exists a morphism τ which differs from σ for a symbol occurring in α, but nevertheless satisfies τ(α)=σ(α); if there is no such τ then σ is called unambiguous. Motivated by the recent initial paper on the ambiguity of morphisms, we introduce the definition of a so-called segmented morphism σ_n, which, for any , maps every symbol in an infinite alphabet onto a word that consists of n distinct factors in , where and are different letters. For every n, we consider the set U(σ_n) of those finite strings over an infinite alphabet with respect to which σ_n is unambiguous, and we comprehensively describe its relation to any U(σ_m), m≠n.Thus, our work features the first approach to a characterisation of sets of strings with respect to which certain fixed morphisms are unambiguous, and it leads to fairly counter-intuitive insights into the relations between such sets. Furthermore, it shows that, among the widely used homogeneous morphisms, most segmented morphisms are optimal in terms of being unambiguous for a preferably large set of strings. Finally, our paper yields several major improvements of crucial techniques previously used for research on the ambiguity of morphisms.     

A Jordan–Brouwer Separation Theorem for Polyhedral Pseudomanifolds

The Jordan Curve Theorem referring to a simple closed curve in the plane has a particularly simple proof in the case that the curve is polygonal, called the “raindrop proof”. We generalize the notion of a simple closed polygon to that of a polyhedral (d−1)-pseudomanifold (d≥2) and prove a Jordan–Brouwer Separation Theorem for such a manifold embedded in ℝ ^d . As a by-product, we get bounds on the polygonal diameter of the interior and exterior of such a manifold which are almost tight. This puts the result within the frame of computational geometry. The research of Y.S. Kupitz was partially supported by the Landau Center at the Mathematics Institute of the Hebrew University of Jerusalem (supported by Minerva Foundation, Germany), and by Deutsche Forschungsgemeinschaft.     

Semi-equivelar maps on the torus and the Klein bottle are Archimedean

If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map $X$ on the torus is a quotient of an Archimedean tiling on the plane then the map $X$ is semi-equivelar. We show that each semi-equivelar map on the torus or on the Klein bottle is a quotient of an Archimedean tiling on the plane.Vertex-transitive maps are semi-equivelar maps. We know that four types of semi-equivelar maps on the torus are always vertex-transitive and there are examples of other seven types of semi-equivelar maps which are not vertex-transitive. We show that the number of $\mathrm{Aut}\left(Y\right)$-orbits of vertices for any semi-equivelar map $Y$ on the torus is at most six. In fact, the number of orbits is at most three except one type of semi-equivelar maps. Our bounds on the number of orbits are sharp.