The goal of this short paper to advertise the method of gauge transformations (aka holographic reduction, reparametrization) that is well-known in statistical physics and computer science, but less known in combinatorics. As an application of it we give a new proof of a theorem of A. Schrijver asserting that the number of Eulerian orientations of a $d$--regular graph on $n$ vertices with even $d$ is at least $\left(\frac{\binom{d}{d/2}}{2^{d/2}}\right)^n$. We also show that a $d$--regular graph with even $d$ has always at least as many Eulerian orientations as $(d/2)$--regular subgraphs.

中文翻译：

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计数受度约束的子图和方向

这篇短文的目标是宣传在统计物理学和计算机科学中广为人知但在组合学中鲜为人知的规范变换方法（又名全息缩减、重新参数化）。作为对它的应用，我们给出了 A. Schrijver 定理的新证明，断言 $d$--正则图在 $n$ 顶点上的欧拉方向数至少为 $\left( \frac{\binom{d}{d/2}}{2^{d/2}}\right)^n$。我们还表明，即使是 $d$ 的 $d$--正则图始终至少具有与 $(d/2)$--正则子图一样多的欧拉方向。