相关论文: A sharp threshold for random graphs with a monochr…

Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic connected components does one need in order to cover its vertex set? This natural question is a well-known essentially equivalent formulation of the classical Ryser's…

Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the…

Kierstead, Szemer\'edi, and Trotter showed that a graph with at most $\lfloor r/(2n)\rfloor^n$ vertices such that each ball of radius $r$ in it is $c$-colorable should have chromatic number at most $n(c-1)+1$. We show that this estimate is…

Given a graph $F$ and an integer $r \ge 2$, a partition $\widehat{F}$ of the edge set of $F$ into at most $r$ classes, and a graph $G$, define $c_{r, \widehat{F}}(G)$ as the number of $r$-colorings of the edges of $G$ that do not contain a…

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…

We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski…

Inspired by the majority colorings and C-colorings, we introduce and study the majority C-coloring of graphs. In such a vertex coloring, every vertex shares its color with at least half of its neighbors. The maximum number of colors that…

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…

When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and…

For graphs $G_0$, $G_1$ and $G_2$, write $G_0\longmapsto(G_1, G_2)$ if each red-blue-edge-coloring of $G_0$ yields a red $G_1$ or a blue $G_2$. The Ramsey number $r(G_1, G_2)$ is the minimum number $n$ such that the complete graph…

The $\mathcal{D}$-process is a single player game in which the player is initially presented the empty graph on $n$ vertices. In each step, a subset of edges $X$ is independently sampled according to a distribution $\mathcal{D}$. The player…

In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that…

Let $\{G_n: n\geq 1\}$ be a sequence of simple graphs. Suppose $G_n$ has $m_n$ edges and each vertex of $G_n$ is colored independently and uniformly at random with $c_n$ colors. Recently, Bhattacharya, Diaconis and Mukherjee (2013) proved…

We say that $G \to (F,H)$ if, in every edge colouring $c: E(G) \to \{1,2\}$, we can find either a $1$-coloured copy of $F$ or a $2$-coloured copy of $H$. The well-known Kohayakawa--Kreuter conjecture states that the threshold for the…

Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is called {\em totally $F$-decomposable} if for {\em every} linear combination of the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is a…

Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow…

We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges…

The size-Ramsey number of a graph $F$ is the smallest number of edges in a graph $G$ with the Ramsey property for $F$, that is, with the property that any 2-colouring of the edges of $G$ contains a monochromatic copy of $F$. We prove that…

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$,…

We construct a hereditary class of triangle-free graphs with unbounded chromatic number, in which every non-trivial graph either contains a pair of non-adjacent twins or has an edgeless vertex cutset of size at most two. This answers in the…