The number of cycles in a graph containing any fixed edge and also containing all vertices of odd degree is odd if and only if all vertices have even degree. If all vertices have even degree this is a theorem of Shunichi Toida. If all vertices have odd degree it is Andrew Thomason's extension of Smith's theorem.

中文翻译：

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包含所有奇数度顶点的循环

当且仅当所有顶点具有偶数阶时，包含任何固定边并且还包含奇数阶的所有顶点的图中的循环数为奇数。如果所有顶点都具有偶数阶数，则这是Shunichi Toida的一个定理。如果所有顶点的奇数度都是安德鲁·托马森对史密斯定理的扩展。