Related papers: Clique colouring of binomial random graphs
A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to…
What is the minimum number of edges that have to be added to the random graph $G=G_{n,0.5}$ in order to increase its chromatic number $\chi=\chi(G)$ by one percent ? One possibility is to add all missing edges on a set of $1.01 \chi$…
The Colouring problem is that of deciding, given a graph $G$ and an integer $k$, whether $G$ admits a (proper) $k$-colouring. For all graphs $H$ up to five vertices, we classify the computational complexity of Colouring for…
A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using…
For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different colors.…
Biclique-colouring is a colouring of the vertices of a graph in such a way that no maximal complete bipartite subgraph with at least one edge is monochromatic. We show that it is coNP-complete to check whether a given function that…
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…
Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the…
We prove for k at most 10, that every graph of chromatic number k with a unique k-coloring admits a clique minor of order k.
We discuss the minimal number of vertices in a graph with a large chromatic number such that each ball of a fixed radius in it has a small chromatic number. It is shown that for every graph $G$ on $\sim((n+rc)/(c+rc))^{r+1}$ vertices such…
The harmonious chromatic number of a graph $G$ is the minimum number of colors that can be assigned to the vertices of $G$ in a proper way such that any two distinct edges have different color pairs. This paper gives various results on…
Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We determine the rate of growth of of the sequence ${Q(n,\lceil rn \rceil)}_{n=1}^\infty$ for any fixed $0<r\leq 1$. We also give a better…
Given d \in (0,infty) let k_d be the smallest integer k such that d < 2k\log k. We prove that the chromatic number of a random graph G(n,d/n) is either k_d or k_d+1 almost surely.
In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs…
The chromatic number $\chi(G)$ of a graph $G$ is defined as the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. The chromatic number of the dense random graph $G \sim G(n,p)$…
In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed,…
Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges…
The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree…
The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…
An acyclic r-coloring of a directed graph G=(V,E) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the…