Related papers: A graphon counter example

We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are…

We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a…

In this paper, we point out that the definition of weak tracial approximation can be improved and strengthened. An example of weak tracial approximation is also provided.

Suppose a finite, unweighted, combinatorial graph $G = (V,E)$ is the union of several (degree-)regular graphs which are then additionally connected with a few additional edges. $G$ will then have only a small number of vertices $v \in V$…

A general method for constructing sharply $k$-arc-transitive digraphs, i.e. digraphs that are $k$-arc-transitive but not $(k+1)$-arc-transitive, is presented. Using our method it is possible to construct both finite and infinite examples.…

We study the geometry of some proper 4-colorings of the vertices of sphere triangulations with degree sequence 6,...,6,2,2,2. Such triangulations are the simplest examples which have non-negative combinatorial curvature. The examples we…

Let $p$ be a real number greater number greater than one. Suppose that a graph $G$ of bounded degree is quasi-isometric with a Riemannian manifold $M$ with certain properties. Under these conditions we will show that the $p$-harmonic…

We present new counterexamples, which provide stronger limitations to sums-differences statements than were previously known. The main idea is to consider non-uniform probability measures.

We construct a H\"older continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with…

We disprove the characterisation of graphs with equal independence and annihilation number by Larson and Pepper (2011). Series of counterexamples with arbitrary number of vertices, arbitrary number of components, arbitrary large…

Graphene may be a fascinating material but at this time there exists no experimental or theoretical evidence that it can improve performance of practical nonlinear optical devices.

We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these…

We prove that for any toric ideal of a graph the degree of any element of Graver basis is bounded above by an exponential function of the maximal degree of a circuit.

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…

Assuming four strongly compact cardinals, it is consistent that all entries in Cicho\'n's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} <…

The asymptotic variety of a counterexample of Pinchuk type to the strong real Jacobian conjecture is explicitly described by low degree polynomials.

We generalize the horofunction compactification to maps that are not distance functions. As an application we define a horofunction counterpart to the Teichm\"uller distance, and discuss its properties.

The paper presents a counterexample to the Hodge conjecture.

We give an Ore-type condition sufficient for a graph G to have a spanning tree with small degrees and with few leaves.

We introduce a family of forcing notions that are helpful in showing that certain graphs do not have countable colourings of (additive) Borel class alpha. We construct graphs that are ''weakly minimal'' for such colourings.