Related papers: On the chromatic number of a random hypergraph

The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Pralat noted that around p \approx n^{-1/2} the clique chromatic number…

In the paper we state and prove theorem describing the upper bound on number of the graphs that have fixed number of vertices |V| and can be colored with the fixed number of n colors. The bound relates both numbers using power of 2, while…

Given independent random points $X_1,...,X_n\in\eR^d$ with common probability distribution $\nu$, and a positive distance $r=r(n)>0$, we construct a random geometric graph $G_n$ with vertex set $\{1,...,n\}$ where distinct $i$ and $j$ are…

We study the problem of constructing a (near) random proper $q$-colouring of a simple k-uniform hypergraph with n vertices and maximum degree \Delta. (Proper in that no edge is mono-coloured and simple in that two edges have maximum…

Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $\chi(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e \in E $ contains at least two vertices of…

Given positive integers $n\ge 2k$, the {\it Kneser graph} $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$-element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical…

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…

Circular chromatic number, $\chi_c$ is a natural generalization of chromatic number. It is known that it is \NP-hard to determine whether or not an arbitrary graph $G$ satisfies $\chi(G) = \chi_c(G)$. In this paper we prove that this…

For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if every hyperedge colouring with $k$ colours of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous…

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of $G$ are…

Let $G$ be a $k$ - connected ($k \geq 2$) graph of order $n$. If $\chi(G) \geq n - k$, then $G$ is Hamiltonian or $K_k \vee (K_k^c \cup K_{n - 2k})$ with $n \geq 2 k + 1$, where $\chi(G)$ is the chromatic number of the graph $G$.

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…

Complete colorings have the property that any two color classes has at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers comes from complete colorings, with some additional requirement. In…

It was conjectured by Ohba and confirmed recently by Noel et al. that, for any graph $G$, if $|V(G)|\le 2\chi(G)+1$ then $\chi_l(G)=\chi(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We show that…

We present here random distributions on $(D+1)$-edge-colored, bipartite graphs with a fixed number of vertices $2p$. These graphs are dual to $D$-dimensional orientable colored complexes. We investigate the behavior of quantities related to…

An $r$-edge coloring of a graph or hypergraph $G=(V,E)$ is a map $c:E\to \{0, \dots, r-1\}$. Extending results of Rado and answering questions of Rado, Gy\'arf\'as and S\'ark\"ozy we prove that (1.) the vertex set of every $r$-edge colored…

Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijver graphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromatic number of random Kneser graphs $\KG_{n,k}(\rho)$ and proved that, in many cases, the…

An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph…

A linearly ordered (LO) $k$-colouring of an $r$-uniform hypergraph assigns an integer from $\{1, \ldots, k \}$ to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for $r=3$, if two vertices…

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…