Related papers: Restricted Color $n$-color Compositions
In most studies related to colouring of graphs and perhaps in the study of other invariants and variants of graphs, the restrictions of non-triviality and connectedness are placed upon graphs. For the introduction to comp\^{o}nent\u{a}…
Carlitz-compositions follow the restrictions of neighbouring parts $\sigma_{i-1}\neq\sigma_{i}$. The recently introduced Arndt-compositions have to satisfy $\sigma_{2i-1}>\sigma_{2i}$. The two concepts are combined to new and exciting…
The work is devoted to one of the variations of the Hadwiger--Nelson problem on the chromatic number of the plane. In this formulation one needs to find for arbitrarily small $\varepsilon$ the least possible number of colors needed to color…
Existing approaches for color-concept association typically rely on query-based image referencing, and color extraction from image references. However, these approaches are effective only for common concepts, and are vulnerable to unstable…
A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs…
Many complex systems and datasets are characterized by multiway interactions of different categories, and can be modeled as edge-colored hypergraphs. We focus on clustering such datasets using the NP-hard edge-colored clustering problem,…
Alon et al. introduced the concept of non-repetitive colourings of graphs. Here we address some questions regarding non-repetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be…
Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\mathfrak{sl}_{n+1})$, $U_q(\mathfrak{sp}_{2n})$ and…
We start by building up some theory to state Wagner's Theorem, and then prove it using Kuratowski's Theorem, a proof of which is found in Diester (2000). Following this, we establish some connections between the chromatic number of a graph…
In our previous paper, we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the…
We study a class of combinatorial objects that we call "decorated trees". These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and…
Let G be an n-vertex graph with list-chromatic number $\chi_\ell$. Suppose each vertex of G is assigned a list of t colors. Albertson, Grossman, and Haas conjecture that at least $t n / {\chi_\ell}$ vertices can be colored from these lists.…
This paper addresses the combinatorial structure of $m$-colored mutation classes. We provide an explicit and purely combinatorial description of the $m$-colored quivers that arise within the $m$-colored mutation class of a quiver of type…
We look at colourings of $r$-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge $E$ in an $r$-uniform hypergraph $H$ whose vertices are coloured is the partition of…
Many combinatorial problems can be formulated as "Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?". An example of such a question is: given two k-colourings of a graph, can I transform the…
We study compositions of a positive integer $n$ in which the occurrence of even parts larger than a fixed threshold $k$ is controlled. More precisely, for each composition $m=(m_1,\dots,m_r)$ we consider the number of even parts strictly…
We introduce a notion of compatibility between constraint encoding and compositional structure. Phrased in the language of category theory, it is given by a "composable constraint encoding". We show that every composable constraint encoding…
Edge-matching problems, also called edge matching puzzles, are abstractions of placement problems with neighborhood conditions. Pieces with colored edges have to be placed on a board such that adjacent edges have the same color. The problem…
The origin of the colours of quarks has been explored and the number of colours equal to three has been derived from the fractal properties suggested in the statistical model.The quark gluon coupling constant has been reproduced and the…
Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…