Listing 6-Cycles in Sparse Graphs
Virginia Vassilevska Williams, Alek Westover

TL;DR
This paper presents a new algorithm for listing 6-cycles in sparse graphs with improved running time, extending previous methods and analyzing cycle counts using supersaturation lemmas.
Contribution
It introduces an $ ilde{O}(m^{1.6}+t)$ time algorithm for listing 6-cycles in sparse graphs, improving upon prior algorithms and providing new analytical techniques.
Findings
Achieved faster listing algorithm with $ ilde{O}(m^{1.6}+t)$ complexity.
Developed a supersaturation lemma relating cycle counts to bipartite graph structure.
Provided a simplified analysis of existing cycle detection algorithms.
Abstract
This work considers the problem of output-sensitive listing of occurrences of -cycles for fixed constant in an undirected host graph with edges and -cycles. Recent work of Jin and Xu (and independently Abboud, Khoury, Leibowitz, and Safier) [STOC 2023] gives an time algorithm for listing -cycles, and recent work by Jin, Vassilevska Williams and Zhou [SOSA 2024] gives an time algorithm for listing -cycles in node graphs. We focus on resolving the next natural question: obtaining listing algorithms for -cycles in the sparse setting, i.e., in terms of rather than . Previously, the best known result here is the better of Jin, Vassilevska Williams and Zhou's algorithm and Alon, Yuster and Zwick's algorithm. We give an algorithm for listing -cycles with running time…
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