# Finding and counting given length cycles

```@article{Alon1997FindingAC,
title={Finding and counting given length cycles},
author={Noga Alon and Raphael Yuster and Uri Zwick},
journal={Algorithmica},
year={1997},
volume={17},
pages={209-223}
}```
• Published 1 March 1997
• Mathematics, Computer Science
• Algorithmica
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.
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## References

SHOWING 1-10 OF 22 REFERENCES

• Computer Science, Mathematics
ESA
• 1994
An assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs improve upon various previously known results.
• Mathematics, Computer Science
SIAM J. Discret. Math.
• 1994
Efficient algorithms for finding even cycles in undirected graphs and finding the shortest even cycle in an Undirected graph G=(V, E) in O(V2) time.
• Mathematics
Inf. Process. Lett.
• 2000
Two algorithms for listing all simplicial vertices of a graph running in time O (n α ) and O (e 2α/(α+1) )= O ( e 1.41) are given, and it is shown that counting the number of K 4 's in a graph can be done in timeO (e ( α+1)/2 ) .
• Computer Science
JACM
• 1983
Smallest-last vertex ordering and prlonty search are utdlzed to show for any graph G = (IT, E) that the set of all connected subgraphs maxunal with respect to their minimum degree can be determined
• Mathematics
Networks
• 1995
An O(n f m) algorithm for recognizing a fixed subgraph H with flower number f within a graph G with n vertices and m edges is presented, which matches the best algorithms known for recognizing small paths, cycles, and cliques.
• Mathematics
STOC '94
• 1994
A novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V,E), which can be derandomized using families of perfect hash functions.
• Mathematics
SIAM J. Comput.
• 1985
A new simple strategy into edge-searching of a graph, which is useful to the various subgraph listing problems, is introduced, and an upper bound on \$a(G)\$ is established for a graph \$G:a (G) \leqq \lceil (2m + n)^{1/2} \rceil \$, where n is the number of vertices in G.