# Cycles

### From G-designs

Relevant articles: _{[1]},
_{[2]},
_{[3]},
_{[4]},
_{[5]}.

## Contents[hide] |

# Cycles

The graphs considered in this section are * cycles*. The cycle with vertices is denoted by .

## Spectrum Results

The spectrum problem for -cycles is completely solved for all , see _{[2]} and _{[4]}.

**Theorem 1**
_{[2]}, _{[4]}

*Let . There exists a -design of order if and only if*

*or ;**is odd; and**.*

## Notes

- -designs are -designs.
- -designs are Steiner triple systems.

## References

- ↑ Adams, P., Bryant, D., and Buchanan, M.
*A survey on the existence of G-designs*, J. Combin. Des.**16**, 373–410 (2008). - ↑
^{2.0}^{2.1}^{2.2}Alspach, B. and Gavlas, H.*Cycle decompositions of K_n and K_n-I*, J. Combin. Theory Ser. B,**81**, 77–99 (2001). - ↑ Hoffman, D. G., Lindner, C. C., and Rodger, C. A.
*On the construction of odd cycle systems*, J. Graph Theory,**13**, 417–426 (1989). - ↑
^{4.0}^{4.1}^{4.2}Šajna, M.*Cycle decompositions. III. Complete graphs and fixed length cycles*, J. Combin. Des.**10**, 27–78 (2002). - ↑ Sotteau, D.
*Decomposition of K_m,n (K^^\ast _m,n) into cycles (circuits) of length 2k*, J. Combin. Theory Ser. B,**30**, 75–81 (1981).