# Stars

### From G-designs

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

## Contents[hide] |

# Stars

The * star on vertices*, denoted by , consists of a vertex of degree , its neighbours, and the edges joining to its neighbours. Note that stars are trees. Also note that a star is the complete bipartite graph .

## Spectrum Results

The spectrum problem for stars was completely settled by Yamamoto et al in _{[2]} and independently by Tarsi in _{[1]}.

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

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

*or ; and**.*

## Notes

- Sometimes is used to denote a star with edges, rather than vertices, for example in
_{[3]}.

## References

- ↑
^{1.0}^{1.1}Tarsi, M.*Decomposition of complete multigraphs into stars*, Discrete Math.**26**, 273–278 (1979). - ↑
^{2.0}^{2.1}^{2.2}Yamamoto, S., Ikeda, H., Shige-eda, S., Ushio, K., and Hamada, N.*On claw-decomposition of complete graphs and complete bigraphs*, Hiroshima Math. J.**5**, 33–42 (1975). - ↑ Adams, P., Bryant, D., and Buchanan, M.
*A survey on the existence of G-designs*, J. Combin. Des.**16**, 373–410 (2008).