In an earlier paper the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositions of odd vertex complete graphs to Hamiltonian cycles admit such an orientation. These conditions are further generalized and some necessary conditions are given as well.

中文翻译：

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将传递锦标赛划分为同构有向图

在较早的一篇论文中，前两位作者已经表明，自补图总是可以以这样的方式定向，即定向版本与其同构定向补的并集给出了一个可传递的锦标赛。我们研究了将该定理推广到将完整图分解为三个或更多同构图的可能性。我们发现，何时可能出现具有相似属性的方向的完整表征似乎难以捉摸。然而，我们给出了充分的条件来推广先前的定理，并且还暗示奇顶点完全图到哈密顿圈的分解承认这样的方向。这些条件被进一步概括，并给出了一些必要条件。