Listing 6-Cycles

Listing copies of small subgraphs (such as triangles, \(4\)-cycles, small cliques) in the input graph is an important and well-studied problem in algorithmic graph theory. In this paper, we give a simple algorithm that lists \(t\) (non-induced) \(6\)-cycles in an \(n\)-node undirected graph in \(\tilde O(n^2+t)\) time. This nearly matches the fastest known algorithm for detecting a \(6\)-cycle in \(O(n^2)\) time by Yuster and Zwick (1997). Previously, a folklore \(O(n^2+t)\)-time algorithm was known for the task of listing \(4\)-cycles.