Related papers: Qualitative representations of chromatic algebras
We study quantum analogs of graph colorings and chromatic number. Initially defined via an interactive protocol, quantum colorings can also be viewed as a natural operator relaxation of graph coloring. Since there is no known algorithm for…
Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of…
An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the…
Motivic characteristic classes of possibly singular algebraic varieties are homology class versions of motivic characteristics, not classes in the so-called motivic (co)homology. This paper is a survey on them with more emphasis on…
Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A…
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory combines these two fields and is concerned with Ramsey-type questions about certain…
We introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a…
The chromatic polynomials are studied by several authors and have important applications in different frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the…
We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial…
We investigate Fair and Tolerant (FAT) graph colorings, a coloring framework in which each vertex is allowed to share its color with a prescribed fraction of its neighbors, while the remaining neighbors are required to be distributed evenly…
Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number…
For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing…
Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all 3 colours. What happens for more colours: if we…
The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction…
A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions…
We study the color patterns that, for $n$ sufficiently large, are unavoidable in $2$-colorings of the edges of a complete graph $K_n$ with respect to $\min \{e(R), e(B)\}$, where $e(R)$ and $e(B)$ are the numbers of red and, respectively,…
The {\em chromatic gap} is the difference between the chromatic number and the clique number of a graph. Here we investigate $\gap(n)$, the maximum chromatic gap over graphs on $n$ vertices. Can the extremal graphs be explored? While…
We use a well known concept of proper vertex colouring of a graph to introduce the construction of a chromatic completion graph and its related parameter, the chromatic completion number of a graph. We then give the chromatic completion…
Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you…