Listing 4-Cycles
Amir Abboud, Seri Khoury, Oree Leibowitz, and Ron Safier

TL;DR
This paper introduces an efficient algorithm for listing all 4-cycles in a graph with a time complexity that surpasses previous bounds for triangle listing, and it is conditionally tight based on recent complexity conjectures.
Contribution
The paper presents a novel algorithm for listing all 4-cycles in a graph with a tight time complexity, separating it from triangle listing complexities.
Findings
Algorithm lists all 4-cycles in ( ilde{O}( ext{min}(n^2,m^{4/3})+t)) time.
The bound is conditionally tight based on recent lower bounds related to the 3-SUM conjecture.
The work distinguishes 4-cycle listing complexity from triangle listing, showing different computational bounds.
Abstract
In this note we present an algorithm that lists all -cycles in a graph in time where is their number. Notably, this separates -cycle listing from triangle-listing, since the latter has a lower bound under the -SUM Conjecture. Our upper bound is conditionally tight because (1) is the best known bound for detecting if the graph has any -cycle, and (2) it matches a recent -SUM lower bound for enumeration algorithms. The latter lower bound was proved very recently by Abboud, Bringmann, and Fischer [arXiv, 2022] and independently by Jin and Xu [arXiv, 2022]. In an independent work, Jin and Xu [arXiv, 2022] also present an algorithm with the same time bound.
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Taxonomy
TopicsLimits and Structures in Graph Theory
