We consider space-efficient algorithms and conditional time lower bounds for finding cycles and walks in graphs. We give a reduction that connects the running time of undirected 2k-cycle to finding directed odd cycles, s-t connectivity in directed graphs, and Max-3-SAT. For example, we show that if 2k-cycle on O(n)-edge graphs can be solved in O(n^(1.5-ε)) time for some ε>0 then, a 2^(n(1-ε')) time algorithm exists for Max-3-SAT for some ε'>0. Additionally, we give a tight combinatorial lower bound for 2k-cycle detection, specifically when k is odd, of m^{2k/(k+1) +o(1)} given the Combinatorial k-Clique Hypothesis.

On the algorithms side, we present a randomized algorithm for directed s-t connectivity using O(lg(n)^2) space and O(n^{lg(n)/2 + o(lg(n))}) expected time, giving a time improvement over Savitch’s famous algorithm, which takes at least n^{lg(n) - o(lg(n))} time. Under the conjecture that every O(lg(n)^2)-space algorithm for directed s-t connectivity requires n^Ω(lg(n)) time, we show that undirected 2k-cycle in O(lg(n)) space requires n^Ω(lg(k)) time.