Cycle decomposition for the permutations of an infinite set

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Cycle Decomposition ◽  
Infinite Set ◽  
Cycle Decompositions

We present two cycle decompositions for the permutations of an infinite set.

scholarly journals Cycle intersection graphs and minimum decycling sets of even graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050027
Author(s):  
Michael Cary
Keyword(s):  
Upper Bound ◽  
Optimization Problem ◽  
Intersection Graph ◽  
Cycle Decomposition ◽  
Intersection Graphs ◽  
Decycling Number ◽  
Cycle Rank ◽  
Cycle Graph ◽  
Cycle Decompositions

We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be significantly better when an even graph admits a cycle decomposition in which any two cycles intersect in at most one vertex. Links between the cycle rank of the cycle intersection graph of an even graph and the decycling number of the even graph itself are found. The problem of choosing an ideal cycle decomposition is addressed and is presented as an optimization problem over the space of cycle decompositions of even graphs, and we conjecture that the upper bound for the decycling number of even graphs presented in this paper is best possible.

scholarly journals Cycle Decompositions of Complete Digraphs

10.37236/8219 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
A. C. Burgess ◽  
P. Danziger ◽  
M. T. Javed
Keyword(s):  
Directed Graph ◽  
Necessary Conditions ◽  
Complete Solution ◽  
Directed Cycle ◽  
Cycle Decomposition ◽  
Alpha Beta ◽  
Large Cycle ◽  
Cycle Decompositions

In this paper, we consider the problem of decomposing the complete directed graph $K_n^*$ into cycles of given lengths.  We consider general necessary conditions for a directed cycle decomposition of $K_n^*$ into $t$ cycles of lengths $m_1, m_2, \ldots, m_t$  to exist and and provide a powerful construction for creating such decompositions in the case where there is one 'large' cycle. Finally, we give a complete solution in the case when there are exactly three cycles of lengths $\alpha, \beta, \gamma \neq 2$. Somewhat surprisingly, the general necessary conditions turn out not to be sufficient in this case.  In particular, when $\gamma=n$, $\alpha+\beta > n+2$ and $\alpha+\beta \equiv n$ (mod 4), $K_n^*$ is not decomposable.

scholarly journals Some Gregarious Cycle Decompositions of Complete Equipartite Graphs

10.37236/224 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Benjamin R. Smith
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cycle Decomposition ◽  
Multipartite Graph ◽  
Necessary And Sufficient ◽  
Cycle Decompositions ◽  
Equipartite Graphs

A $k$-cycle decomposition of a multipartite graph $G$ is said to be gregarious if each $k$-cycle in the decomposition intersects $k$ distinct partite sets of $G$. In this paper we prove necessary and sufficient conditions for the existence of such a decomposition in the case where $G$ is the complete equipartite graph, having $n$ parts of size $m$, and either $n\equiv 0,1\pmod{k}$, or $k$ is odd and $m\equiv 0\pmod{k}$. As a consequence, we prove necessary and sufficient conditions for decomposing complete equipartite graphs into gregarious cycles of prime length.

scholarly journals A Note on On-Line Ramsey Numbers for Some Paths

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 735
Author(s):  
Tomasz Dzido ◽  
Renata Zakrzewska
Keyword(s):  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
On Line ◽  
Infinite Set ◽  
Do So

We consider the important generalisation of Ramsey numbers, namely on-line Ramsey numbers. It is easiest to understand them by considering a game between two players, a Builder and Painter, on an infinite set of vertices. In each round, the Builder joins two non-adjacent vertices with an edge, and the Painter colors the edge red or blue. An on-line Ramsey number r˜(G,H) is the minimum number of rounds it takes the Builder to force the Painter to create a red copy of graph G or a blue copy of graph H, assuming that both the Builder and Painter play perfectly. The Painter’s goal is to resist to do so for as long as possible. In this paper, we consider the case where G is a path P4 and H is a path P10 or P11.

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

An Infinite Set of Heron Triangles with Two Rational Medians

1997 ◽  
Vol 104 (2) ◽  
pp. 107-115 ◽  
Author(s):  
Ralph H. Buchholz ◽  
Randall L. Rathbun
Keyword(s):  
Infinite Set

scholarly journals Time and Life in the Relational Universe: Prolegomena to an Integral Paradigm of Natural Philosophy

Philosophies ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 30 ◽  
Author(s):  
Abir Igamberdiev
Keyword(s):  
Probability Function ◽  
Natural Philosophy ◽  
Natural World ◽  
Spatiotemporal Structure ◽  
Internal Logic ◽  
Decision Making System ◽  
Relational Domains ◽  
Relational Order ◽  
Infinite Set ◽  
The Universe

Relational ideas for our description of the natural world can be traced to the concept of Anaxagoras on the multiplicity of basic particles, later called “homoiomeroi” by Aristotle, that constitute the Universe and have the same nature as the whole world. Leibniz viewed the Universe as an infinite set of embodied logical essences called monads, which possess inner view, compute their own programs and perform mathematical transformations of their qualities, independently of all other monads. In this paradigm, space appears as a relational order of co-existences and time as a relational order of sequences. The relational paradigm was recognized in physics as a dependence of the spatiotemporal structure and its actualization on the observer. In the foundations of mathematics, the basic logical principles are united with the basic geometrical principles that are generic to the unfolding of internal logic. These principles appear as universal topological structures (“geometric atoms”) shaping the world. The decision-making system performs internal quantum reduction which is described by external observers via the probability function. In biology, individual systems operate as separate relational domains. The wave function superposition is restricted within a single domain and does not expand outside it, which corresponds to the statement of Leibniz that “monads have no windows”.

XLIX.—Generalizations of a Problem of Pillai

1949 ◽  
Vol 62 (4) ◽  
pp. 460-469
Author(s):  
L. Mirsky
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Main Concern ◽  
Positive Integers ◽  
Infinite Set

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.

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