Related papers: Forcing among patterns with no block structure

We consider the stability of periodic map with period-$2$ in linear fractional difference equations where the function is $f(x)=ax$ at even times and $f(x)=bx$ at odd times. The stability of such a map for an integer order map depends on…

In a finite undirected simple graph, a chordless cycle is an induced subgraph which is a cycle. A graph is called cyclically orientable if it admits an orientation in which every chordless cycle is cyclically oriented. We propose an…

An even hole is an induced chordless cycle of even length at least four. A diamond is an induced subgraph isomorphic to K_4-e. We show that graphs without even holes and without diamonds can be decomposed via clique-separators into graphs…

Let $q$ be an odd prime power. Let $f\in \mathbb{F}_q[x]$ be a polynomial having degree at least $2$, $a\in \mathbb{F}_q$, and denote by $f^n$ the $n$-th iteration of $f$. Let $\chi$ be the quadratic character of $\mathbb{F}_q$, and…

A graph is near-bipartite if its vertex set can be partitioned into an independent set and a set which induces a forest. In this paper, planar graphs without cycles of length from 4 to 7 are shown to be near-bipartite.

Recently Lin, Wang and Zhou have proved that every $3$-connected nonbipartite graph of minimum degree at least $k$ with $k\ge 6$ and order at least $k+2$ contains $k$ cycles of consecutive lengths. They also conjecture that this result is…

We analyze the stopping-time and cycle structure of the normalized Collatz iteration. Using a recursive description of admissible binary sequences, we show that every integer $m \equiv 3 \pmod{4}$ arises uniquely and derive new bounds for…

For a graph $G$, let $\sigma_{2}(G)$ be the minimum degree sum of two non-adjacent vertices in $G$. A chord of a cycle in a graph $G$ is an edge of $G$ joining two non-consecutive vertices of the cycle. In this paper, we prove the following…

A meander system is a union of two arc systems that represent non-crossing pairings of the set $[2n] = \{1, \ldots, 2n\}$ in the upper and lower half-plane. In this paper, we consider random meander systems. We show that for a class of…

We consider the cycle structure of a random permutation $\sigma$ chosen uniformly from the symmetric group, subject to the constraint that $\sigma$ does not contain cycles of length exceeding $r.$ We prove that under suitable conditions the…

The goal of this article is to prove the comparison theorem between algebraic and topological nearby cycles of a morphism without slopes. We prove in particular that for a family of holomorphic functions without slopes, if we iterate…

This work shows a study about the structure of the cycles contained in a Minimal Strong Digraph (MSD). The structure of a given cycle is determined by the strongly connected components (or strong components, SCs) that appear after…

Consider a sequence of LexBFS vertex orderings {\sigma}1, {\sigma}2, . . . where each ordering {\sigma}i is used to break ties for {\sigma}i+1. Since the total number of vertex orderings of a finite graph is finite, this sequence must end…

Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to…

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect…

For all positive integers $s$ and $t$ exceeding one, a matroid $M$ on $n$ elements is {\em nearly $(s, t)$-cyclic} if there is a cyclic ordering $\sigma$ of its ground set such that every $s-1$ consecutive elements of $\sigma$ are contained…

For sufficiently nice families of semigroups and monoids, the structure theorem for sets of length states that the length set of any sufficiently large element is an arithmetic sequence with some values omitted near the ends. In this paper,…

In 1975, Erd\H{o}s asked the following natural question: What is the maximum number of edges that an $n$-vertex graph can have without containing a cycle with all diagonals? Erd\H{o}s observed that the upper bound $O(n^{5/3})$ holds since…

Dean conjectured that for each integer $k \ge 3$, every graph with minimum degree at least $k$ has a cycle whose length is divisible by $k$; this conjecture is known to be true for all $k\neq 5$. For $k\in\{3,4\}$, stronger statements are…

We prove that there exist functions $f,g:\mathbb{N}\to\mathbb{N}$ such that for all nonnegative integers $k$ and $d$, for every graph $G$, either $G$ contains $k$ cycles such that vertices of different cycles have distance greater than $d$…