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Let Kv for the complete graph with v vertices , and when v is even , Kv-F for the complete graph with v vertices minus a 1 - factor . Kv ( or Kv-F) can be split into a circle lengths of m1, m2, ... , mt circle C1, C2, ..., Ct is the necessary conditions : (1) 3 ≤ mi ≤ v (1 ≤ i ≤ t); (2) v ≡ 1 (mod2) ( or v ≡ 0 (mod2)) ; (3) m1 m2 ... mt = (?) ( or m1 m2 ... mt = (?)). Alspach conjecture in 1981 : necessary condition is sufficient . difficult to solve this problem , so that last for 23 years, although many people have made ??efforts to this conjecture , but the result was few and largely confined circle collection contains only two kinds of long cycle length . in this paper, spin , recursion , construction and other methods to solve the when the length of the loop when the set is { 3,6,8 } , Alspach conjecture is correct (v = 34,46 exception ) . article is divided into three parts, the first part of the prior knowledge , including the text used in the cited marks and related reasonable second part is the main body of articles , will all complete graph by order of the parity of the two types of v to prove Alspach conjecture . v is even Shiyou modulo 24 ( 3,6,8 LCM ) of 12 categories, with each category there is a corresponding PBD or GDD but will order a complete graph split into smaller complete graph ; v odd, most of the recursive method is the application of an even number from v deduced from the case , and the remaining the situation will be a complete graph split into two complete graph with a complete bipartite graph , while the use of the structure and scaling , making proved to be more concise , which is the innovation of this paper . third part appendices, including a second v In the recursive part outside the structure obtained by the direct decomposition of laps .
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