Related papers: Set maps, umbral calculus, and the chromatic polyn…

For each infinite cardinal k, the set of algebraic hypergraphs having chromatic number no larger than k is decidable.

There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have…

We consider some problems concerning the maximum number of (strong) dominating sets in a regular graph, and their weighted analogues. Our primary tool is Shearer's entropy lemma. These techniques extend to a reasonably broad class of graph…

We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…

Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…

We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $\lambda$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference…

Recently, big data techniques such as machine learning and topological data analysis have made their way to theoretical mathematics. Motivated by the recent work with polynomial invariants for knots, we use manifold learning and topological…

The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed…

We define a $P$-compelling coloring as a proper coloring of the vertices of a graph such that every subset consisting of one vertex of each color has property $P$. The $P$-compelling chromatic number is the minimum number of colors in such…

For any cubic graph in a closed orientable surface and a perfect matching, the Penrose-Kauffman polynomial is a sum of chromatic polynomials of a collection of associated graphs. A knot-theoretic perspective affords elementary proofs of old…

In this paper we study the set chromatic number of a random graph $G(n,p)$ for a wide range of $p=p(n)$. We show that the set chromatic number, as a function of $p$, forms an intriguing zigzag shape.

We define an infinite set of families of graphs, which we call $p$-wheels and denote $(Wh)^{(p)}_n$, that generalize the wheel ($p=1$) and biwheel ($p=2$) graphs. The chromatic polynomial for $(Wh)^{(p)}_n$ is calculated, and remarkably…

We show that the existence of a universal countably chromatic graph of size $\aleph_1$ together with the failure of continuum hypothesis is consistent. The proof is a forcing iteration of strongly proper ccc posets. The construction works…

Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…

The expansion of bivariate polynomials is well-understood for sets with a linear-sized product set. In contrast, not much is known for sets with small sumset. In this work, we provide expansion bounds for polynomials of the form $f(x, y) =…

For $a,b\in\mathbb{N}_0$, we consider $(an+b)$-color compositions of a positive integer $\nu$ for which each part of size $n$ admits $an+b$ colors. We study these compositions from the enumerative point of view and give a formula for the…

In this paper we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. The results include approximation…

The theory of algebraic extensions of Banach algebras is well established, and there are many constructions which yield interesting extensions. In particular, Cole's method for extending uniform algebras by adding square roots of functions…

A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated integrals. Both classes lead to sets of…

Using the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that…