We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for some $k$) instead of cycles (graphs with all degrees even). We give an almost-exponential lower-bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower-bound for planar graphs. We conjecture that any bridgeless cubic graph has at least $2^{n/2-1}$ circuit double covers and we show an infinite class of graphs for which this bound is tight.

翻译：我们研究循环双覆盖猜想的计数版本。我们讨论了为何计数电路（即同构于某$k$的$C_k$的图）比计数环（所有顶点度数为偶数的图）更有意义。对于亏格嵌入表示性至少为4的图，我们给出了一个近乎指数的下界。我们还证明了平面图的指数下界。我们猜想任何无桥三次图至少具有$2^{n/2-1}$个电路双覆盖，并展示了一个无限图类使得该下界是紧的。