scholarly journals Balanced directed cycle designs based on cyclic groups

1995 ◽  
Vol 58 (2) ◽  
pp. 210-218 ◽  
Author(s):  
Chaufah Nilrat ◽  
Cheryl E. Praeger
Keyword(s):  
Directed Graph ◽  
Cyclic Group ◽  
Complete Classification ◽  
Directed Cycle ◽  
Group Of Automorphisms ◽  
Edge Disjoint ◽  
Cyclic Groups ◽  
Directed Cycles

AbstractA balanced directed cycle design with parameters (υ, k, 1), sometimes called a (υ, k, 1)-design, is a decomposition of the complete directed graph into edge disjoint directed cycles of length k. A complete classification is given of (υ, k, 1)-designs admitting the holomorph {øa, b: x ↦ ax + b∣ a, b ∈ Zυ, (a, υ1) = 1} of the cyclic group Zυ as a group of automorphisms. In particular it is shown that such a design exists if and ony if one of (a) k = 2, (b) p ≡ 1 (mod k) for each prime p dividing υ, or (c) k is the least prime dividing υ, k2 does not divide υ, and p ≡ 1 (mod k) for each prime p < k dividing υ.

scholarly journals Directed Graphs Without Short Cycles

2009 ◽  
Vol 19 (2) ◽  
pp. 285-301 ◽  
Author(s):  
JACOB FOX ◽  
PETER KEEVASH ◽  
BENNY SUDAKOV
Keyword(s):  
Directed Graph ◽  
Absolute Constant ◽  
Simple Structure ◽  
Directed Graphs ◽  
Constant Factor ◽  
Directed Cycle ◽  
Strong Component ◽  
Feedback Arc Set ◽  
Directed Cycles

For a directed graph G without loops or parallel edges, let β(G) denote the size of the smallest feedback arc set, i.e., the smallest subset X ⊂ E(G) such that G ∖ X has no directed cycles. Let γ(G) be the number of unordered pairs of vertices of G which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least r ≥ 4 satisfies β(G) ≤ cγ(G)/r2, where c is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour and Sullivan.This result can also be used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed 0 < θ < 1/2 and sufficiently large n, if G is a digraph with n vertices and β(G) ≥ θn2, then for any 0 ≤ m ≤ θn − o(n) it contains a directed cycle whose length is between m and m + 6θ−1/2. Moreover, there is a constant C such that either G contains directed cycles of every length between C and θn − o(n) or it is close to a digraph G′ with a simple structure: every strong component of G′ is periodic. These results are also tight up to the constant factors.

scholarly journals On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
K. Mageshwaran ◽  
G. Kalaimurugan ◽  
Bussakorn Hammachukiattikul ◽  
Vediyappan Govindan ◽  
Ismail Naci Cangul
Keyword(s):  
Cyclic Group ◽  
Upper Bound ◽  
Labeling Index ◽  
Positive Difference ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
The Difference ◽  
Radio Labeling

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

Conditions for the Separability of Objects in Two-Dimensional Velocity Fields

1992 ◽  
Vol 35 (4) ◽  
pp. 484-491
Author(s):  
Stephan Foldes
Keyword(s):  
Velocity Field ◽  
Directed Graph ◽  
Velocity Fields ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Jordan Curves ◽  
Directed Cycles

AbstractWe consider the directed graph representing the obstruction relation between objects moving along the streamlines of a two-dimensional velocity field. A collection of objects is sequentially separable if and only if the corresponding graph has no directed cycles. A sufficient condition for this is the permeability of closed Jordan curves.

scholarly journals Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

On the sumset of atoms in cyclic groups

2014 ◽  
Vol 10 (06) ◽  
pp. 1355-1363 ◽  
Author(s):  
Cui-Fang Sun ◽  
Quan-Hui Yang
Keyword(s):  
Cyclic Group ◽  
Exact Formula ◽  
Cyclic Groups

For a cyclic group G and a ∈ G, we define 〈a〉 as the subgroup generated by a and the atom of a as the set of all elements generating 〈a〉. In this paper, given t (t ≥ 2) atoms, we obtain an exact formula for the number of representations of each element in the sumset of these t atoms. We also show explicitly which atoms are part of the union which constitutes the sumset of t given atoms. The case t = 2 was recently obtained by Sander and Sander.

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

scholarly journals Generalized Fibonacci groups H(r,n,s) that are connected labelled oriented graph groups

2019 ◽  
Vol 22 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Gerald Williams
Keyword(s):  
Cyclic Group ◽  
Betti Numbers ◽  
Oriented Graph ◽  
Complete Classification ◽  
Circulant Matrices ◽  
Fundamental Groups ◽  
Infinite Cyclic Group ◽  
Torus Knot ◽  
Graph Groups

Abstract The class of connected Labelled Oriented Graph (LOG) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in {S^{4}} , and so contains all knot groups. We investigate when Campbell and Robertson’s generalized Fibonacci groups {H(r,n,s)} are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups {H(r,n,s)} that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that {H(r,n,s)} is a 2-generator knot group if and only if it is a torus knot group.

scholarly journals Mathematical Properties of the Hyperbolicity of Circulant Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Juan C. Hernández ◽  
José M. Rodríguez ◽  
José M. Sigarreta
Keyword(s):  
Cyclic Group ◽  
Metric Space ◽  
Geodesic Triangle ◽  
Group Of Automorphisms ◽  
Mathematical Properties ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Circulant Networks ◽  
Hyperbolicity Constant

IfXis a geodesic metric space andx1,x2,x3∈X, ageodesic triangle  T={x1,x2,x3}is the union of the three geodesics[x1x2],[x2x3], and[x3x1]inX. The spaceXisδ-hyperbolic(in the Gromov sense) if any side ofTis contained in aδ-neighborhood of the union of the two other sides, for every geodesic triangleTinX. The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.

scholarly journals Homotopy Formulas for Cyclic Groups Acting on Rings

2008 ◽  
Vol 51 (1) ◽  
pp. 81-85
Author(s):  
Christian Kassel
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Cohomology Groups ◽  
Joint Work ◽  
Cyclic Groups ◽  
A Finite Group

AbstractThe positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

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