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Relevant articles: [1].


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).


  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).