# Paths

Relevant articles: [1].

# Paths

The path with $m$ vertices is denoted $P_m$. Note that paths are trees.

## Spectrum Results

The spectrum problem for paths was completely settled by Tarsi in [1].

Theorem 1 [1]

Let $m\geq 2$. There exists a $P_m$-design of order $n$ if and only if

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

## References

1. 1.0 1.1 1.2 Tarsi, M. Decomposition of a complete multigraph into simple paths: nonbalanced handcuffed designs, J. Combin. Theory Ser. A, 34, 60–70 (1983).