Holey Cycle Design And ( 5 ,8 ) - Cycle Systems R S

Combinatorial mathematics is a branch of mathematics on how to arrange (or allocate) theobject of discretion under the given constraints.This subject can date back to 2200 BC when ourcountry was still in Dayu times-a ?ood control era in ancient China,yet it developed so slowlythat only untill the invention of computers in 1940s did it begin to develop and grow with anunprecedented speed,and had been widely applied in more and more aspects such as computernetworks, digital communications, experimental design and modern business management. Theprogress of modern science and technology and the rapid development of computer and infor-mation related disciplines make the content of study become ever richer in fields of modulardesign, graph theory, network theory and coding design in combinatorial mathematics. Atthe same time,many questions of important theoretical significance and prospects have beenraised,such as the famous Alspach's conjecture which is the object of this article.Let Kv be a complete graph with v vertices, and K_v - F be a complete graph with a1-factor removed . The necessary conditions for the existence of a decomposition of Kv (orK_v - F ) into cycles C₁,C₂,···Ct of lengths m₁,m₂,···m_t are:(1)(2)v≡1(mod 2),(orv≡0(mod 2));(3)The question of these necessary conditions are sufficient was asked by Alspach.We de-scribe a method which, in certain circumstances, may be used to prove that the necessary con-ditions are sufficient. In this paper, we prove that these necessary conditions are sufficient whenm_i∈{5,8},for i = 1,2,···t.A decomposition of Kv(orK_v - F) into cycles all of the same length m which partitionthe edge set of Kv(orK_v - F) is usually called an m-cycle system of K_v(or K_v - F). Althoughthe existence problem for m-cycle systems ofKv(orK_v - F)has been completely solved. theexistence problem for m-cycle systems ofKv\ Ku(or(K_v-F₁)\ (K_u-F₂)) does not completelysolve . Especially when u,v is an even, only the following two cases have been resolved. D.Bryant solves the situation of when m = 5, Adams,Bryant and Khodkar solves the situationwhen m=3,whereas this article solves when u,v are both even , m≡0(mod 2)and4≤m≤14.The main content of this article are as following:Chaper one, introduce some concepts used in this article, sign and main theorems . Chapter two, from the necessary conditions that the existence problem for m-cycle systemsof the holey graph G ,and with the application of ideas in Number Theory, the conditions ofsystems of the original holey graph G has come out by analysis.Then ,skillfully split the holeygraph G into a number of known graphs or to concretly construct systems of the original holeygraph G with the ideas in Set Theory and the method of Difference Sets. Finally,with theapplication of Theorem 2.2,the existence problem for m-cycle systems of(Kv ?F1)\ (Ku?F2)has been completely solved. when m is even, and m∈{4, 6, 8, 10, 12, 14}.Chaper 3, we have already known the results to the existence problem for m-cycle systemsofK_v\ K_u,when m≤14. With the above results and the knowledge of graph theory,G hasbeen skillfully split into several sub-graphs and a recursive structure has been made by usingthe existing m-cycle systems in known graphs,constructing a small order of the general graphG of (5^r, 8^s)-cycle systems. With the the results to the existence problem for 5-cycle systemsofK_v \ K_u and the application of Theorem 1.2.2, a finding point by the usage of FunctionTheory, finally, on the basis of the already constructed small order of graph G of (5r, 8s)?cyclesystems and with the usage of combinatorial design theory and mathematical induction, it isproved that when v is odd, and m_i∈{5, 8},i = 1, 2,···t, Alspach's conjecture is available.Chaper 4, with the already obtained results to the existence problem for 8-cycle systems of(K_v-F₁)\ (K_u-F₂) and to the existence problem for 5-cycle systems of(K_v-F₁)\ (K_u-F₂), the graph G has been skillfully split into several sub-graph and a recursive structure has beenmade by using the result that there is m-cycle systems in known graphs, the exsiting of thegeneral small graphs G of (5^r, 8^s)- cycle systems has been proved. Finally,with all the resultsabove and mathematical induction,it has proved that when v is even, and m_i∈{5, 8},i =1, 2,···t, Alspach's conjecture is available.