Related papers: On variants of conflict-free-coloring for hypergra…
We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where…
We study dynamic conflict-free colorings in the plane, where the goal is to maintain a conflict-free coloring (CF-coloring for short) under insertions and deletions. - First we consider CF-colorings of a set $S$ of unit squares with respect…
Fix $r \ge 2$ and a collection of $r$-uniform hypergraphs $\cH$. What is the minimum number of edges in an $\cH$-free $r$-uniform hypergraph with chromatic number greater than $k$. We investigate this question for various $\cH$. Our results…
The paper deals with an extremal problem concerning equitable colorings of uniform hyper\-graph. Recall that a vertex coloring of a hypergraph $H$ is called proper if there are no monochro-matic edges under this coloring. A hypergraph is…
A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic…
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…
Vertex coloring and multicoloring of graphs are a well known subject in graph theory, as well as their applications. In vertex multicoloring, each vertex is assigned some subset of a given set of colors. Here we propose a new kind of vertex…
In graph coloring problems, the goal is to assign a positive integer color to each vertex of an input graph such that adjacent vertices do not receive the same color assignment. For classic graph coloring, the goal is to minimize the…
We study a graph coloring problem that is otherwise easy but becomes quite non-trivial in the one-pass streaming model. In contrast to previous graph coloring problems in streaming that try to find an assignment of colors to vertices, our…
For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a…
Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be…
For any c >= 2, a c-strong coloring of the hypergraph G is an assignment of colors to the vertices of G such that for every edge e of G, the vertices of e are colored by at least min{c,|e|} distinct colors. The hypergraph G is…
A balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the…
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…
We consider the following game, played on a $k$-uniform hypergraph $H$. There are $q$ colors available and two players take it in turns to color vertices. A partial coloring is proper if no edge is mono-chromatic. One player, A, wishes to…
We study rainbow-free colourings of $k$-uniform hypergraphs; that is, colourings that use $k$ colours but with the property that no hyperedge attains all colours. We show that $p^*=(k-1)(\ln n)/n$ is the threshold function for the existence…
A triangle in a hypergraph is a collection of distinct vertices u,v,w and distinct edges e,f,g with u,v \in e, v,w \in f, w,u \in g, and \{u,v,w\} \cap e \cap f \cap g=\emptyset. The i-degree of a vertex in a hypergraph is the number of…
The conflict-free chromatic index of a graph $G$ is the minimum number of colours in an edge colouring of $G$ such that the neighbourhood of every edge contains a colour appearing exactly once. Its vertex analogue is the conflict-free…
Let $\mathcal{H}$ be a hypergraph of maximal vertex degree $\Delta$, such that each its hyperedge contains at least $\delta$ vertices. Let $k=\lceil\frac{2\Delta}{\delta}\rceil$. We prove that (i) The hypergraph $\mathcal{H}$ admits proper…
In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will…