 
Summary: Finding and counting given length cycles
Noga Alon
Raphael Yuster
Uri Zwick
Abstract
We present an assortment of methods for finding and counting simple cycles of a given length in
directed and undirected graphs. Most of the bounds obtained depend solely on the number of
edges in the graph in question, and not on the number of vertices. The bounds obtained improve
upon various previously known results.
1 Introduction
The main contribution of this paper is a collection of new bounds on the complexity of finding
simple cycles of length exactly k, where k 3 is a fixed integer, in a directed or an undirected graph
G = (V, E). These bounds are of the form O(Ek ) or of the form O(Ek ·d(G)k ), where d(G) is the
degeneracy of the graph (see below). The bounds improve upon previously known bounds when the
graph in question is relatively sparse or relatively degenerate.
We let Ck stand for a simple cycle of length k. When considering directed graphs, a Ck is assumed
to be directed. We show that a Ck in a directed or undirected graph G = (V, E), if one exists, can
be found in O(E2 2
k ) time, if k is even, and in O(E2 2
k+1 ) time, if k is odd. For finding triangles
