Related papers: Chromatic thresholds for linear equations and recu…

Let $n\ge 1$, $r\ge 2$, and $s\ge 0$ be integers and ${\cal P}=\{P_1,\dots, P_l\}$ be a partition of $[n]=\{1,\dots, n\}$ with $|P_i|\le r$ for $i=1,\dots, l$. Also, let $\cal F$ be a family of non-empty subsets of $[n]$. The $r$-uniform…

By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $\Delta$ has chromatic number at most $(C+o(1))\Delta/\log \Delta$ for some universal constant $C > 0$. Using the entropy compression method, Molloy proved that one…

In 2006, Collins and Trenk obtained a general sharp upper bound for the distinguishing chromatic number of a connected graph. Inspired by Catlin's combinatorial techniques from 1978, we establish improved upper bounds for classes of…

A graph class is $\chi$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erd\H{o}s conjectured that intersection graphs of straight-line segments in the plane are…

Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey…

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the…

Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden…

We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge…

A system of linear equations $L$ is common over $\mathbb{F}_p$ if, as $n\to\infty$, any 2-coloring of $\mathbb{F}_p^n$ gives asymptotically at least as many monochromatic solutions to $L$ as a random 2-coloring. The notion of common linear…

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $\chi_{\rm pcf}^h(G)$ denote the minimum $k$ such that…

Let $G$ be a graph with maximum degree $\Delta$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours…

Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to…

In this paper, we are interested in some problems related to chromatic number and clique number for the class of $(P_5,K_5-e)$-free graphs, and prove the following. $(a)$ If $G$ is a connected ($P_5,K_5-e$)-free graph with $\omega(G)\geq…

A triangle in a hypergraph $\mathcal{H}$ is a set of three distinct edges $e, f, g\in\mathcal{H}$ and three distinct vertices $u, v, w\in V(\mathcal{H})$ such that $\{u, v\}\subseteq e$, $\{v, w\}\subseteq f$, $\{w, u\}\subseteq g$ and…

We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$ or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, our result…

Hoffman proved that a graph $G$ with eigenvalues $\mu_1 \ge \ldots \ge \mu_n$ and chromatic number $\chi(G)$ satisfies: \[ \chi \ge 1 + \kappa \] where $\kappa$ is the smallest integer such that \[ \mu_1 + \sum_{i=1}^{\kappa} \mu_{n+1-i}…

The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$.…

Let R be a commutative ring with unity (not necessarily finite). The ring R consider as a simple graph whose vertices are the elements of R with two distinct vertices x and y are adjacent if xy=0 in R, where 0 is the zero element of R. In…

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property.…

We study a generalisation of Vizing's theorem, where the goal is to simultaneously colour the edges of graphs $G_1,\dots,G_k$ with few colours. We obtain asymptotically optimal bounds for the required number of colours in terms of the…