Abstract
Let C k denote a cycle of length k and let S k denote a star with k edges. As usual K n denotes the complete graph on n vertices. In this paper we investigate decomposition of K n into C l ’s and S k ’s, and give some necessary or sufficient conditions for such a decomposition to exist. In particular, we give a complete solution to the problem in the case l = k = 4 as follows: For any nonnegative integers p and q and any positive integer n, there exists a decomposition of K n into p copies of C 4 and q copies of S 4 if and only if \({4(p + q)={n \choose 2}, q\ne 1}\) if n is odd, and \({q\geq max\{3, \lceil{\frac{n}{4}\rceil}\}}\) if n is even.
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This work was supported by the National Science Council of Taiwan (NSC 97-2115-M-003-006).
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Shyu, TW. Decomposition of Complete Graphs into Cycles and Stars. Graphs and Combinatorics 29, 301–313 (2013). https://doi.org/10.1007/s00373-011-1105-3
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DOI: https://doi.org/10.1007/s00373-011-1105-3