Related papers: Generalized 3x + 1 Mappings : counting cycles

We establish closed-form expansions for the number of colorings of a path or cycle on n vertices with colors from 1,...,x such that adjacent vertices are colored differently or with colors from y+1,...x.

We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s \ge 2$ and sufficiently large $n$. Moreover, for $s\in\{2,3\}$ and any integer $n$ we obtain the maximum number of cycles of…

Collatz Conjecture (also known as Ulam's conjecture and 3x+1 problem) concerns the behavior of the iterates of a particular function on natural numbers. A number of generalizations of the conjecture have been subjected to extensive study.…

We consider the multiplicity of limit cycles that appear when a hyperbolic polycycle is perturbed. We prove, in particular, that if such unfolding happens in generic finite-parameter families, the multiplicity of every new limit cycle does…

We give a new proof of the result that if f and g are transcendental entire functions, then the composite function f(g) has infinitely many fixed points. The method yields a number of generalization of this result. In particular, it extends…

In this note we provide a two-dimensional family of smooth minimal threefolds of general type with canonical map of degree 96, improving the previous known bound of 72.

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i)…

It was shown by Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2022 that every induced $2$-edge path in a vertex-transitive graph closes to an induced cycle. Similar results were obtained for 3-edge paths closing to cycles in…

Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small…

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions:…

Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism…

After a brief discussion of the history of the problem, we propose a generalization of the map colouring problem to higher dimensions.

We prove that a linear growth graph has finitely many horofunctions. This provides a short and simple proof that any finitely generated infinite group of linear growth is virtually cyclic.

A theorem on the existence of exactly $N$ limit cycles around a critical point for the Lienard system $\ddot{x}+f(x) \dot{x}+g(x) =0$ is proved. An alogrithm on the determination of a desired number of limit cycles for this system has been…

In this paper cyclic codes are established with respect to the Mannheim metric over some finite rings by using Gaussian integers and the decoding algorithm for these codes is given.

We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the…

We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family…

This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.

In this paper we describe the dynamics of certain rational maps of the form $k \cdot (x+x^{-1})$ over finite fields of odd characteristic.