We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kuhn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament $T$. There is a natural lower bound for this number in terms of the degree sequence of $T$ and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.

中文翻译：

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将锦标赛分解为路径

我们考虑了1976年Alspach，Mason和Pullman提出的Kelly猜想的一般化。Kelly的猜想指出，每场常规比赛都有汉密尔顿循环的边分解，这在大型比赛中得到了Kuhn和Osthus的证明。Alspach，Mason和Pullman的猜想要求在一般比赛$ T $的路径分解中需要的最小路径数。根据$ T $的度数顺序，此数字有一个自然的下限，并且可以推测该下限对于偶数顺序的锦标赛是正确的。猜想的几乎所有情况都是公开的，我们证明了其中许多情况。