Listing 4-Cycles

Abstract

In this note we present an algorithm that lists all $4$-cycles in a graph in time $\tilde{O}(\min (n^2,m^{4/3})+t)$ where $t$ is their number. Notably, this separates $4$-cycle listing from triangle-listing, since the latter has a $(\min (n^3,m^{3/2})+t{)}^{1-o(1)}$ lower bound under the $3$-SUM Conjecture. Our upper bound is conditionally tight because (1) $O(n^2,m^{4/3})$ is the best known bound for detecting if the graph has any $4$-cycle, and (2) it matches a recent $(\min (n^3,m^{3/2})+t{)}^{1-o(1)}$ $3$-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.