Related papers: Fractional chromatic number of a random subgraph

The determination of the quantum chromatic number of graphs has attracted considerable attention recently. However, there are few families of graphs whose quantum chromatic numbers are determined. A notable exception is the family of…

Given a graph $G$, its Hall ratio $\rho(G)=\max_{H\subseteq G}\frac{|V(H)|}{\alpha(H)}$ forms a natural lower bound on its fractional chromatic number $\chi_f(G)$. A recent line of research studied the fundamental question of whether…

A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is…

The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…

A topological index of a graph $G$ is a real number which is preserved under isomorphism. Extensive studies on certain polynomials related to these topological indices have also been done recently. In a similar way, chromatic versions of…

Hofmman's bound on the chromatic number of a graph states that $\chi \geq 1 - \frac {\lambda_1} {\lambda_n}$. Here we show that the same bound, or slight modifications of it, hold for several graph parameters related to the chromatic…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $d n$ is bounded by a constant depending only on $H$ and $d$.…

We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$ edges has a fractional triangle decomposition, for $n \ge 7$. This is a key ingredient in our proof, given in a companion paper, that every $n$-vertex $2$-coloured…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\Delta(G)>n/3$ has…

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in…

Let $k, d$ ($2d \leq k)$ be two positive integers. We generalize the well studied notions of $(k,d)$-colorings and of the circular chromatic number $\chi_c$ to signed graphs. This implies a new notion of colorings of signed graphs, and the…

In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model $\Gnd$ for $d=o(n^{1/5})$ is concentrated in two consecutive values, thus extending a previous result of Achlioptas…

A {\em chromatic root} is a root of the chromatic polynomial of a graph. While the real chromatic roots have been extensively studied and well understood, little is known about the {\em real parts} of chromatic roots. It is not difficult 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 $1.01 \chi$…

Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if and only if x + y is in A. Then, almost surely as N tends to infinity, the chromatic…

A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex…

Complete colorings have the property that any two color classes has at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers comes from complete colorings, with some additional requirement. In…

Chromatic polynomials are important objects in graph theory and statistical physics, but as a result of computational difficulties, their study is limited to graphs that are small, highly structured, or very sparse. We have devised and…

A large body of research in graph theory concerns the induced subgraphs of graphs with large chromatic number, and especially which induced cycles must occur. In this paper, we unify and substantially extend results from a number of…

Let $G$ be a graph on $n$ vertices and let $\mathcal{L}_k$ be an arbitrary function that assigns each vertex in $G$ a list of $k$ colours. Then $G$ is $\mathcal{L}_k$-list colourable if there exists a proper colouring of the vertices of $G$…