# Cycles

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

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# Cycles

The graphs considered in this section are cycles. The cycle with $m$ vertices is denoted by $C_m$.

## Spectrum Results

The spectrum problem for $m$-cycles is completely solved for all $m$, see [2] and [4].

Theorem 1 [2], [4]

Let $m\geq 3$. There exists a $C_m$-design of order $n$ if and only if

• $n=1$ or $n\geq m$;
• $n$ is odd; and
• $n(n-1)\equiv 0 \,({\rm mod \ }2m)$.

## Notes

• $C_3$-designs are $K_3$-designs.
• $C_3$-designs are Steiner triple systems.

## References

1. Adams, P., Bryant, D., and Buchanan, M. A survey on the existence of G-designs, J. Combin. Des. 16, 373–410 (2008).
2. 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).
3. 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. 4.0 4.1 4.2 Šajna, M. Cycle decompositions. III. Complete graphs and fixed length cycles, J. Combin. Des. 10, 27–78 (2002).
5. 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).