Listing All Fixed-Length Simple Cycles in Sparse Graphs in Optimal Time

The degeneracy of an n-vertex graph G is the smallest number k such that every subgraph of G contains a vertex of degree at most k. We present an algorithm for enumerating all simple cycles of length p in an n-order k-degenerate graph running in time O(n([p/2]) k([p/2])). We then show that this algorithm is worst-case output size optimal by proving a T(n([p/2]) k([p/2])) bound on the maximal number of simple p-length cycles in these graphs. Our results also apply to induced (chordless) cycles.