Related papers: On Andreae's Ubiquity Conjecture

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph ${\mathcal G}_{n,1/2}$ is, in particular, 3-random-like, this can be viewed as a weak…

We characterise the slices of the category of graphs that are algebraically universal in terms of the structure of the slicing graph. In particular, we show that algebraic universality is obtained if, and only if, the slicing graph contains…

A subgraph-universal graph/a topological minor-universal graph in a class of graphs $\mathcal{G}$ is a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as a subgraph/topological minor. We prove that the class $\mathcal{P}$…

The Kohayakawa-Nagle-R\"odl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important…

Let G be a simple finite graph such that each vertex has an integer value and different vertices have different values. Let S be a finite non-empty set of primes. We call G an S-graph if any two vertices are connected by an edge if and only…

A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges…

For a family $\mathcal{F}$ of graphs, a graph $G$ is called \emph{$\mathcal{F}$-universal} if $G$ contains every graph in $\mathcal{F}$ as a subgraph. Let $\mathcal{F}_n(d)$ be the family of all graphs on $n$ vertices with maximum degree at…

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each $k$, there is a finite obstruction set $\mathcal{O}_k$ of graphs such that…

A random rooted graph is said to be sofic if it is the Benjamini-Schramm limit of a sequence of finite graphs. Given any finite graph $H$, we prove that every one-ended, unimodular random rooted graph that does not have H as a minor must be…

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most…

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our…

Given a connected locally finite simplicial graph $ G$ with vertex set $V$, the combinatorial Laplacian $\Delta_G \colon \R^V \to \R^V$ is defined on the space of all real-valued functions on $V$. We prove that $\Delta_G$ is surjective if…

Answering a question of H\"aggkvist and Scott, Verstra\"ete proved that every sufficiently large graph with average degree at least $k^2+19k+10$ contains $k$ vertex-disjoint cycles of consecutive even lengths. He further conjectured that…

The celebrated Erdos-Hajnal conjecture states that for every $n$-vertex undirected graph $H$ there exists $\eps(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or an independent set of size…

A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive.…

A graph is universally $k$-edge-weightable if for every $k$-element set $Q\subset\mathbb{R}$, it admits a proper $Q$-edge weighting. The settled 1-2-3 conjecture implies that for any arithmetic progression $\{a,b,c\}$, every nice regular…

If $G$ is a simple graph and $\rho\in[0,1]$, the reliability $R_G(\rho)$ is the probability of $G$ being connected after each of its edges is removed independently with probability $\rho$. A simple graph $G$ is a \emph{uniformly most…

An {\em antimagic labeling} of a graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges…

For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n…

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing…