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Decomposition of complete graphs into paths and stars. (English) Zbl 1219.05146
Summary: Let \(P_{k+1}\) denote a path of length \(k\) and let \(S_{k+1}\) denote a star with \(k\) edges. As usual \(K_n\) denotes the complete graph on \(n\) vertices. In this paper we investigate the decomposition of \(K_n\) 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 \(K_n\) into \(p\) copies of \(P_{4}\) and \(q\) copies of \(S_{4}\) if and only if \(n\geq 6\) and \(3(p+q) = \binom n2\).
Theorem B. Let \(p\) and \(q\) be nonnegative integers, let \(n\) and \(k\) be positive integers such that \(n\geq 4k\) and \(k(p+q) = \binom n2\), and let one of the following conditions hold:
(1)
\(k\) is even and \(p \geq \frac k2\),
(2)
\(k\) is odd and \(p\geq k\).
Then there exists a decomposition of \(K_n\) into \(p\) copies of \(P_{k+1}\) and \(q\) copies of \(S_{k+1}\).

MSC:
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C38 Paths and cycles
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