Enumerating Small Cycles

In a seminal result of Yuster and Zwick, they showed that for any fixed $k$, the even cycle $C_{2k}$ can be detected in an $n$-vertex graph in time $O(n^2)$. For $4$-cycles, a folklore algorithm extends to listing: for any $t$, we can list $t$ different $4$-cycles, if such exist, in $O(n^2+t)$ time. Recently, Jin, Vassilevska-Williams, and Zhou obtained similar bounds for listing $6$-cycles. In this work, we generalize the above to cycles of sizes $8, 10, 12, 14,$ and $16$; we show that for all $k\leq 8$, we can list $t$ distinct $2k$-cycles in $\tilde{O}(n^2+t)$ time. In fact, our algorithm gives enumeration with pre-processing time $\tilde{O}(n^2)$ and delay $\tilde{O}(1)$. Additionally, for any fixed $k$, we present an optimal enumeration (and hence also listing) algorithm for all cycles of size at most $2k$. More generally, for any fixed $k$ and any $3\le i\le \frac{4k}{3}$, we present an algorithm with preprocessing time $\tilde{O}(n^2)$ and delay $\tilde{O}(1)$ that enumerates all cycles of sizes in the range $[i,2k]$.