### Abstract

Let D be the triangle with an attached edge (i.e. D is the "kite", a graph having vertices {a0,a1,a2,a3} and edges {a0,a1}, {a0,a2}, {a1,a2}, {a0,a3}). Bermond and Schönheim [G-decomposition of Kn, where G has four vertices or less, Discrete Math. 19 (1977) 113-120] proved that a kite-design of order n exists if and only if n≡0or1(mod8). Let (W,C) be a nontrivial kite-design of order n≥8, and let V⊂W with |V|=v<n. A path design (V,P) of order v and block size s is embedded into (W,C) if there is an injective mapping f:P→C such that B is an induced subgraph of f(B) for every B∈P. For each n≡0or1(mod8), we determine the spectrum of all integers v such that there is a nontrivial path design of order v and block size 3 embedded into a kite-design of order n.

Original language | English (US) |
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Pages (from-to) | 38-48 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 297 |

Issue number | 1-3 |

DOIs | |

State | Published - Jul 28 2005 |

### Keywords

- Embedding
- Graph design
- Path

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*297*(1-3), 38-48. https://doi.org/10.1016/j.disc.2005.04.014