Related papers: Restricted Color $n$-color Compositions
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
We establish closed-form expansions for the number of colorings of a path or cycle on n vertices with colors from 1,...,x such that adjacent vertices are colored differently or with colors from y+1,...x.
For graph classes $P_1,...,P_k$, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph $G$ can be partitioned into subsets $V_1,...,V_k$ so that $V_j$ induces a graph in the class $P_j$…
In this paper, based on the contributions of Tucker (1983) and Seb{\H{o}} (1992), we generalize the concept of a sequential coloring of a graph to a framework in which the algorithm may use a coloring rule-base obtained from suitable…
It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more…
Nivat's conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps $\mathbb Z^2 \rightarrow \mathcal A$ where $\mathcal A$ is a finite set of symbols. Such configurations are often…
We develop a topological framework in an attempt to generalize the classical colourful Caratheodory theorem by imposing an additional constraint. For that we introduce the notion of zero-avoding complexes and covering criteria for the…
Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded…
We introduce moduli spaces of colored graphs, defined as spaces of non-degenerate metrics on certain families of edge-colored graphs. Apart from fixing the rank and number of legs these families are determined by various conditions on the…
In $2003$, Alladi, Andrews and Berkovich proved a four-parameter partition identity lying beyond a celebrated identity of G\"ollnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this…
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a…
The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are…
A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many…
Interference phenomena are the source of some of the spectacular colors of animals and plants in nature. In some of these systems, the physical structure consists of an ordered array of layers with alternating high and low refractive…
An exactly solvable model is used to investigate the assumptions behind color transparency.
We both survey and extend a new technique from Lu Liu to prove separation theorems between products of Ramsey-type theorems over computable reducibility. We use this technique to show that Ramsey's theorem for $n$-tuples and three colors is…
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
We consider an urn model, whose replacement matrix has all entries nonnegative and is balanced, that is, has constant row sums. We obtain the rates of the counts of balls corresponding to each color for the strong laws to hold. The analysis…
An S-restricted composition of a positive integer n is an ordered partition of n where each summand is drawn from a given subset S of positive integers. There are various problems regarding such compositions which have received attention in…
The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs…