Related papers: Large chromatic number and Ramsey graphs

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture…

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 clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study…

In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3,…

Vertex coloring of a graph $G$ with $n$-colors can be equivalently thought to be a graph homomorphism (edge preserving vertex mapping) of $G$ to the complete graph $K_n$ of order $n$. So, in that sense, the chromatic number $\chi(G)$ of $G$…

The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of…

Problem of finding an optimal upper bound for a chromatic no. of 3K1-free graphs is still open and pretty hard. Here we find a tight chromatic upper bound for {3K1, C5}-free graphs. We prove that if G is {3K1, C5}-free, then the chromatic…

The anti-Ramsey number, $AR(n,G)$, for a graph $G$ and an integer $n\geq|V(G)|$, is defined to be the minimal integer $r$ such that in any edge-colouring of $K_n$ by at least $r$ colours there is a multicoloured copy of $G$, namely, a copy…

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…

Ramsey's theorem, in the version of Erd\H{o}s and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,...,n} contains a monochromatic clique of order 1/2\log n. In this paper, we consider two well-studied…

The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the…

For graphs $H_1,H_2$ by $r^*(H_1,H_2)$ we denote the minimum number of edges in a graph $G$ on $r(H_1,H_2)$ vertices such that $G\to (H_1,H_2)$. We show that for each pair of natural numbers $k,n$, $k\le n$, where $k$ is odd and $n$ is…

We prove that the chromatic number of a circle graph with clique number $\omega$ is at most $7\omega^2$.

The size-Ramsey number $\hat r(G')$ of a graph $G'$ is defined as the smallest integer $m$ so that there exists a graph $G$ with $m$ edges such that every $2$-coloring of the edges of $G$ contains a monochromatic copy of $G'$. Answering a…

One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on $N$ vertices. By the celebrated result of Erd\H{o}s from 1947, the random graph on $N$ vertices with edge…

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…

If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently…

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very…

In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles.

Denote by $R(G_1, G_2, G_3)$ the minimum integer $N$ such that any three-colouring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of a graph $G_i$ coloured with colour $i$ for some $i\in{1,2,3}$. In a…