Listing 4-Cycles
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In this note we present an algorithm that lists all 4-cycles in a graph in time Õ(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.
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