Decomposition of complete graphs into paths and stars

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Let Pk + 1 denote a path of length k and let Sk + 1 denote a star with k edges. As usual Kn denotes the complete graph on n vertices. In this paper we investigate the decomposition of Kn into paths and stars, and prove the following results. Theorem A. Let p and q be nonnegative integers and let n be a positive integer. There exists a decomposition of Kn into p copies of P4 and q copies of S4 if and only if n ≥ 6 and 3 (p + q) = fenced(frac(n, 2)). Theorem B. Let p and q be nonnegative integers, let n and k be positive integers such that n ≥ 4 k and k (p + q) = fenced(frac(n, 2)), and let one of the following conditions hold: (1)k is even and p ≥ frac(k, 2),(2)k is odd and p ≥ k. Then there exists a decomposition of Kn into p copies of Pk + 1 and q copies of Sk + 1.

Original languageEnglish
Pages (from-to)2164-2169
Number of pages6
JournalDiscrete Mathematics
Issue number15-16
Publication statusPublished - 2010 Aug 28



  • Complete graph
  • Decomposition
  • Path
  • Star

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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