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：我们研究了一个循环双覆盖猜想的计数版本。我们讨论了为什么计算电路（即同构于$C_k$的图形，其中$k$为某个整数）而不是循环（所有度数均为偶数的图形）更有趣。我们给出了表现性至少为4的表面嵌入图的几乎指数下界。我们还证明了平面图的指数下界。我们猜测任何无桥三次正则图至少有$2^{n/2-1}$个电路双覆盖，并展示了一个确保该界是紧的无限类图形。