Related papers: When $t$-intersecting hypergraphs admit bounded $c…

A graph $H$ is common if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring. We prove that, given $k,r>0$, there exists a $k$-connected common…

An edge-colored graph $G$ is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by $rc(G)$, is the minimum number of colors needed to…

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…

A proper coloring $c$ of a simple graph $G$ is harmonious if, for every pair of distinct edges $uv,xy\in E(G)$, we have that $\{c(u),c(v)\}\neq \{c(x),c(y)\}$. The harmonious chromatic number of $G$, denoted by $h(G)$, is the least positive…

A hypergraph $\mathcal{H}$ on $n$ vertices and $m$ edges is said to be {\it nearly-intersecting} if every edge of $\mathcal{H}$ intersects all but at most polylogarthmically many (in $m$ and $n$) other edges. Given lists of colors…

We introduce and study conflict-free colourings of $t$-subsets in hypergraphs. In such colourings, one assigns colours to all subsets of vertices of cardinality $t$ such that in any hyperedge of cardinality at least $t$ there is a uniquely…

Let $G$ be an edge-colored graph. We use $e(G)$ and $c(G)$ to denote the number of edges and colors in $G$, respectively. A subgraph $H$ is called rainbow if $c(H)=e(H)$. Li et al. (European J. Combin., 36 (2014), 453-459) proved that every…

A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other…

Given a hypergraph $H$, the conflict-free colouring problem is to colour vertices of $H$ using minimum colours so that each hyperedge in $H$ sees a unique colour. We present a polynomial time reduction from the conflict-free colouring…

We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace…

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…

For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\geq \binom{n+1}{2}+k-1$ for a…

A classical result of Erd\H{o}s and Hajnal claims that for any integers $k, r, g \geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ with chromatic number at least $k$. This implies that there are sparse hypergraphs such that…

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…

A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough…

The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are…

For a proper vertex coloring $c$ of a graph $G$, let $\varphi_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $\chi(H)$ and the number of colors used by $c$ to color $H$. We…

Let $G(V,E)$ be a $k$-uniform hypergraph. A hyperedge $e \in E$ is said to be properly $(r,p)$ colored by an $r$-coloring of vertices in $V$ if $e$ contains vertices of at least $p$ distinct colors in the $r$-coloring. An $r$-coloring of…

This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the $d$-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance $d$ are of a different…

The {\em acyclic chromatic number} of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The {\em acyclic chromatic index} is the analogous graph parameter for edge…