Listing 6-Cycles in Sparse Graphs

TL;DR

The paper studies output-sensitive listing of even cycles C_{2k} in sparse graphs, focusing on the challenging C_6 case. It extends the Dahlgaard–Knudsen–Stoeckel framework for C_{2k} detection by bounding capped k-walks via a layered-bucketing analysis, and it introduces the Unbalanced Supersaturation Conjecture as a guiding principle. The main result is the first improvement over the long-standing O(m^{5/3}+t) bound for C_6 listing, achieving O(m^{1.6}+t) time, with a simplified analytic toolkit and conditional extensions. The work advances understanding of sparse cycle listing, provides a concrete route to generalizing to larger C_{2k}, and supplies methodological tools (simplified analysis, supersaturation-based reasoning, and a computer-assisted LP case split) that could influence both theory and practice in graph algorithms.

Abstract

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 Stöckel'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 Stöckel's $2k$-cycle detection algorithm (with a small polylogarithmic increase in the running time), which is helpful in analyzing our listing algorithm.

Figures (1)

• Figure 1: A layer pattern capturing many capped $3$-walks.

Theorems & Definitions (19)

• Theorem 1
• Theorem 1
• Theorem 1
• Theorem 2
• Conjecture 2
• Theorem 4
• Lemma 5
• Corollary 6
• Lemma 7
• Lemma 8