Related papers: Linear chord diagrams with long chords

A construction sequence for a graph is a listing of the elements of the graph (the set of vertices and edges) such that each edge follows both its endpoints. The construction number of the graph is the number of such sequences. We determine…

How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erd\H{o}s and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$…

The mathematics of musical intervals and scales has been extensively studied. Vastly simplified, our ears seem to prefer intervals whose frequency ratios have small numerator and denominator, such as 2:1 (octave), 3:2 (perfect fifth), 4:3…

The class of closed graphs by a linear ordering on their sets of vertices is investigated. A recent characterization of such a class of graphs is analyzed by using tools from the proper interval graph theory.

In this paper we give for any integer l > 2 a numerical criterion ensuring the existence of a chain of length l of lines through two general points of an irreducible variety X in P^N, involving the degrees and the number of homogeneous…

For a linear code $C$ of length $n$ with dimension $k$ and minimum distance $d$, it is desirable that the quantity $kd/n$ is large. Given an arbitrary field $\mathbb{F}$, we introduce a novel, but elementary, construction that produces a…

It is known that the longest simple path in the divisor graph that uses integers $\leq N$ is of length $\asymp N/\log N$. We study the partitions of $\{1,2,\dots, N\}$ into a minimal number of paths of the divisor graph, and we show that in…

Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a…

We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.

In this paper we discuss the number of regions in a unit circle after drawing $n$ i.i.d. random chords in the circle according to a particular family of distribution. We find that as $n$ goes to infinity, the distribution of the number of…

For every integer $\ell \geq 7$, we give a structural description of the class of graphs whose chordless cycles of length at least 4 all have length $\ell$.

We determine the number of labelled chordal planar graphs with $n$ vertices, which is asymptotically $c_1\cdot n^{-5/2} \gamma^n n!$ for a constant $c_1>0$ and $\gamma \approx 11.89235$. We also determine the number of rooted simple chordal…

For a degree sequence, we define the set of edges that appear in every labeled realization of that sequence as forced, while the edges that appear in none as forbidden. We examine structure of graphs whose degree sequences contain either…

We tackle the problem of constructing increasing-chord graphs spanning point sets. We prove that, for every point set P with n points, there exists an increasing-chord planar graph with O(n) Steiner points spanning P. Further, we prove…

Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece.…

Let $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices…

A chordless cycle (induced cycle) $C$ of a graph is a cycle without any chord, meaning that there is no edge outside the cycle connecting two vertices of the cycle. A chordless path is defined similarly. In this paper, we consider the…

We prove that constant minimum degree already forces cycles with almost linearly many chords. Specifically, every graph $G$ with $\delta(G)\ge C$ contains a cycle of length $\ell\ge 4$ with $\Omega(\ell/\log^{C}\ell)$ chords for some…

We say that two sets $S,T\subset\{1,2,\dots,n\}$ are chord separated if there does not exist a cyclically ordered quadruple $a,b,c,d$ of integers satisfying $a,c\in S-T$ and $b,d\in T-S$. This is a weaker version of Leclerc and Zelevinsky's…

We introduce nominal string diagrams as, string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we…