Related papers: The Lovasz number of random graphs
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of…
We determine the asymptotics of the independence number of the random $d$-regular graph for all $d \ge d_0$. It is highly concentrated, with constant-order fluctuations around $n\alpha_* - c_*\log n$ for explicit constants $\alpha_*(d)$ and…
Let $s\ge2$ and $t\ge2$ be integers. A graph $G$ is $(s,t)$-\emph{splittable} if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $\chi(G[S])\geq s$ and $\chi(G[T])\geq t$. The well-known Erd\H{o}s-Lov\'asz Tihany Conjecture…
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for…
In this paper we solve two problems of Esperet, Kang and Thomasse as well as Li concerning (i) induced bipartite subgraphs in triangle-free graphs and (ii) van der Waerden numbers. Each time random greedy algorithms allow us to go beyond…
The chromatic number $\chi(G)$ of a graph $G$, that is, the smallest number of colors required to color the vertices of $G$ so that no two adjacent vertices are assigned the same color, is a classic and extensively studied parameter. Here…
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…
We study the chromatic number of typical triangle-free graphs with $\Theta \left( n^{3/2} (\log n)^{1/2} \right)$ edges and establish the width of the scaling window for the transitions from $\chi = 3$ to $\chi = 4$ and from $\chi = 4$ to…
We introduce the asymptotic spectrum of graphs and apply the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new dual characterisation of the Shannon capacity of graphs. Elements in the asymptotic spectrum…
The aim of this note is twofold. On the one hand, we present a streamlined version of Molloy's new proof of the bound $\chi(G) \leq (1+o(1))\Delta(G)/\ln \Delta(G)$ for triangle-free graphs $G$, avoiding the technicalities of the entropy…
Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an \emph{$L$-system} is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ such that $|F \cap F'| \in L$ for distinct $F, F'\in \mathcal{F}$. $L$-systems correspond…
We show that the independence number of $ G_{n,p}$ is concentrated on two values if $ n^{-2/3+ \epsilon} < p \le 1$. This result is roughly best possible as an argument of Sah and Sawhney shows that the independence number is not, in…
Let G(n,d) be the random d-regular graph on n vertices. For any integer k exceeding a certain constant k_0 we identify a number d_{k-col} such that G(n,d) is k-colorable w.h.p. if d<d_{k-col} and non-k-colorable w.h.p. if d>d_{k-col}.
We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\leq \log n$,…
Hoffman proved that for a simple graph $G$, the chromatic number $\chi(G)$ obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and $\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$…
We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for…
We show that the Kneser graph of triangulations of a convex $n$-gon has chromatic number $n-2$.
In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most $k$ if it has a fractional coloring in…
Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded…
We investigate vector chromatic number, Lovasz theta of the complement, and quantum chromatic number from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e. that for each of…