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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
##### Keywords:
decomposition; path; star; complete graph
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##### References:
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