A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges and even otherwise. Pontus von Brömssen (né Andersson) showed that the existence of such an automorphism is independent of the orientation and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic even graphs on n vertices is equal to the number of pairwise non-isomorphic tournaments on n vertices. We prove this conjecture using a counting argument with several applications of the Cauchy–Frobenius theorem.

一个图被称为奇数，如果它的边有一个方向，并且自同构反转它的奇数个边的意义，否则。Pontus von Brömssen (né Andersson) 证明了这种自同构的存在与方向无关，并考虑了计算成对非同构偶数图的问题。基于计算证据，他做出了一个相当令人惊讶的猜想，即 n 个顶点上的成对非同构偶数图的数量等于n个顶点上的成对非同构锦标赛的数量。我们使用柯西-弗罗贝尼乌斯定理的几个应用的计数论证来证明这个猜想。