Related papers: Square pegs between two graphs

We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments.…

Let $A$ and $B$ be unital rings. An additive map $T:A\to B$ is called a weighted Jordan homomorphism if $c=T(1)$ is an invertible central element and $cT(x^2) = T(x)^2$ for all $x\in A$. We provide assumptions, which are in particular…

We can view the Lipschitz constant as a height function on the space of maps between two manifolds and ask (as Gromov did nearly 30 years ago) what its ``Morse landscape'' looks like: are there high peaks, deep valleys and mountain passes?…

Let $n\in \Bbb N,$ and let $A,B$ be two rings. An additive map $h: A\to B$ is called n-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $a \in {A}$. Every Jordan homomorphism is an n-Jordan homomorphism, for all $n\geq 2,$ but the converse…

We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least $n$. In the operator case, it was recently…

We study Lefschetz pencils on symplectic four-manifolds via the associated spheres in the moduli spaces of curves, and in particular their intersections with certain natural divisors. An invariant defined from such intersection numbers can…

We prove that for every integer $t\geq 1$, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most $t$ points is $\chi$-bounded. This is essentially the strongest…

We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of…

It was proved by Scott that for every $k\ge2$, there exists a constant $c(k)>0$ such that for every bipartite $n$-vertex graph $G$ without isolated vertices, there exists an induced subgraph $H$ of order at least $c(k)n$ such that…

We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given $\alpha \in (0,1)$, the union of any $n$-vertex graph with minimum degree $\alpha n$ and the binomial random…

Let $S_i$, $i\in I$, be a countable collection of Jordan curves in the extended complex plane $\Sph$ that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated,…

Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized…

By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation…

We compute and provide a detailed description on the Jordan constants of the multiplicative subgroup of quaternion algebras over number fields of small degree. As an application, we determine the Jordan constants of the multiplicative…

For an oriented graph $G$, the oriented discrepancy problem concerns the existence of a spanning subgraph of $G$ with a large imbalance between its forward and backward edge orientations. Freschi and Lo proved the Dirac-type Hamilton cycle…

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a…

Let $A$ and $B$ be complex unital Banach algebras, and let $\varphi, \psi: A \to B$ be surjective mappings. If $A$ is semisimple with an essential socle and $\varphi$ and $\psi$ preserves the invertibility of linear pencils in both…

We give a necessary and sufficient condition so that a pair of disjoint Jordan regions in the sphere can be quasiconformally mapped to a pair of disks. As a consequence, we obtain a simple characterization that involves Lipschitz functions…

Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split…

Jordan isomorphisms of rings are defined by two equations. The first one is the equation of additivity while the second one concerns multiplicativity with respect to the so-called Jordan product. In this paper we present results showing…