This work considers the problem of output-sensitive listing of occurrences of $2k$-cycles for fixed constant $k\geq 2$ in an undirected host graph with $m$ edges and $t$ $2k$-cycles. Recent work of Jin and Xu (and independently Abboud, Khoury, Leibowitz, and Safier) [STOC 2023] gives an $O(m^{4/3}+t)$ time algorithm for listing $4$-cycles, and recent work by Jin, Vassilevska Williams and Zhou [SOSA 2024] gives an $\widetilde{O}(n^2+t)$ time algorithm for listing $6$-cycles in $n$ node graphs. We focus on resolving the next natural question: obtaining listing algorithms for $6$-cycles in the sparse setting, i.e., in terms of $m$ rather than $n$. Previously, the best known result here is the better of Jin, Vassilevska Williams and Zhou's $\widetilde{O}(n^2+t)$ algorithm and Alon, Yuster and Zwick's $O(m^{5/3}+t)$ algorithm. We give an algorithm for listing $6$-cycles with running time $\widetilde{O}(m^{1.6}+t)$. Our algorithm is a natural extension of Dahlgaard, Knudsen and St\"ockel's [STOC 2017] algorithm for detecting a $2k$-cycle. Our main technical contribution is the analysis of the algorithm which involves a type of ``supersaturation'' lemma relating the number of $2k$-cycles in a bipartite graph to the sizes of the parts in the bipartition and the number of edges. We also give a simplified analysis of Dahlgaard, Knudsen and St\"ockel's $2k$-cycle detection algorithm (with a small polylogarithmic increase in the running time), which is helpful in analyzing our listing algorithm.

翻译：本研究探讨在具有 $m$ 条边且包含 $t$ 个 $2k$ 环的无向宿主图中，针对固定常数 $k\geq 2$ 实现输出敏感的 $2k$ 环枚举问题。Jin 和 Xu（以及 Abboud、Khoury、Leibowitz 和 Safier 的独立工作）[STOC 2023] 近期提出了 $4$ 环枚举的 $O(m^{4/3}+t)$ 时间算法；Jin、Vassilevska Williams 和 Zhou [SOSA 2024] 近期提出了在 $n$ 节点图中枚举 $6$ 环的 $\widetilde{O}(n^2+t)$ 时间算法。我们聚焦于解决下一个自然问题：在稀疏场景下（即用 $m$ 而非 $n$ 表示）获得 $6$ 环枚举算法。此前，该问题最著名的结果是 Jin、Vassilevska Williams 和 Zhou 的 $\widetilde{O}(n^2+t)$ 算法与 Alon、Yuster 和 Zwick 的 $O(m^{5/3}+t)$ 算法中的较优者。我们提出了一种运行时间为 $\widetilde{O}(m^{1.6}+t)$ 的 $6$ 环枚举算法。该算法是 Dahlgaard、Knudsen 和 Stöckel [STOC 2017] 提出的 $2k$ 环检测算法的自然扩展。我们的主要技术贡献在于算法分析，其中涉及一类“超饱和”引理，该引理将二分图中 $2k$ 环的数量与二分划分中各部分的规模及边数相关联。我们还对 Dahlgaard、Knudsen 和 Stöckel 的 $2k$ 环检测算法进行了简化分析（运行时间有微小多项式对数级增加），这有助于分析我们的枚举算法。