指数级数量的回路双覆盖
Exponentially Many Circuit Double Covers
AI总结：
研究立方图回路双覆盖的计数版本，通过线性方程组系统，证明2边连通3边可着色立方图至少有2^{n/2 - 1}个回路双覆盖，给出围长至少16的3边连通立方图的较弱指数下界，还刻画了立方图平面性及给出5回路双覆盖的流条件。
AI中文摘要：
塞凯赖什和西摩的回路双覆盖猜想，OpenAI最近宣布了其证明，该猜想称每个无桥图都有一组回路恰好覆盖每条边两次。我们研究立方图此陈述的计数版本，即计算回路双覆盖——覆盖每条边两次的回路（连通2正则子图）集合。我们表明，每个具有n个顶点的2边连通3边可着色立方图至少有2^{n/2 - 1}个回路双覆盖，与我们之前推测的一般下界相匹配。对于每个围长至少为16的3边连通立方图，我们给出了回路双覆盖的较弱指数下界。对于这两个结果，我们使用了OpenAI在其证明中使用的相同线性方程组系统，不过我们提供了额外的组合解释。我们通过该方程组对于任意无处零\(\mathbb Z_2^k\)流的可解性来刻画立方图的平面性。我们给出了一个与存在5回路双覆盖等价的流的条件。
英文摘要：
The cycle double cover conjecture of Szekeres and Seymour, the proof of which was recently announced by OpenAI, states that every bridgeless graph has a collection of cycles covering every edge exactly twice. We study the counting version of this statement for cubic graphs, where we count circuit double covers --- collections of circuits (connected 2-regular subgraphs) covering every edge twice. We show that every 2-edge-connected 3-edge-colorable cubic graph on $n$ vertices has at least $2^{n/2-1}$ circuit double covers, matching our previously conjectured general lower bound. For every 3-edge-connected cubic graph with girth at least 16 we show a weaker exponential lower bound on circuit double covers. For both of these results we use the same system of linear equations used by OpenAI in their proof, however, we provide additional combinatorial interpretation. We characterize planarity of a cubic graph by solvability of this system of equations for arbitrary nowhere-zero $\mathbb Z_2^k$-flow. We give a condition on the flow that is equivalent to existence of a 5-cycle double cover.
