Related papers: Unique-maximum and conflict-free colorings for hyp…
A properly edge-colored graph is a graph with a coloring of its edges such that no vertex is incident to two or more edges of the same color. A subgraph is called rainbow if all its edges have different colors. The problem of finding…
The inclusion relation between simple objects in the plane may be used to define geometric set systems, or hypergraphs. Properties of various types of colorings of these hypergraphs have been the subject of recent investigations, with…
Determining the maximum number of edges in an intersecting hypergraph on a fixed ground set under additional constraints is one of the central topics in extremal combinatorics. In contrast, there are few results on analogous problems…
A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a…
A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies such a coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption…
We investigate the extent to which the $k$-coloring graph $\mathcal{C}_{k}(G)$ uniquely determines the base graph $G$ and the number of colors $k$. The vertices of $\mathcal{C}_{k}(G)$ are the proper $k$-colorings of $G$, and edges connect…
A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number,…
A classic graph coloring problem is to assign colors to vertices of any graph so that distinct colors are assigned to adjacent vertices. Optimal graph coloring colors a graph with a minimum number of colors, which is its chromatic number.…
A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are…
The input of the Maximum Colored Cut problem consists of a graph $G=(V,E)$ with an edge-coloring $c:E\to \{1,2,3,\ldots , p\}$ and a positive integer $k$, and the question is whether $G$ has a nontrivial edge cut using at least $k$ colors.…
We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored…
The list coloring problem is a variant of vertex coloring where a vertex may be colored only a color from a prescribed set. Several applications of vertex coloring are more appropriately modelled as instances of list coloring and thus we…
This paper investigates vertex colorings of graphs such that some rainbow subgraph~$R$ and some monochromatic subgraph $M$ are forbidden. Previous work focussed on the case that $R=M$. Here we consider the more general case, especially the…
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…
The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a…
A proper coloring $\phi$ of $G$ is called a proper conflict-free coloring of $G$ if for every non-isolated vertex $v$ of $G$, there is a color $c$ such that $|\phi^{-1}(c)\cap N_G(v)|=1$. As an analogy to degree-choosability of graphs, the…
In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive…
For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same…
Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring…
A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are…