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Isomorphic star decompositions of multicrowns and the power of cycles. (English) Zbl 0994.05113
A crown $$C_{n,k}$$ for integers $$n,k$$ with $$k\leq n$$ is the graph whose vertex set is $$\{a_1,\dots ,a_n, b_1,\dots ,b_n\}$$ and in which each edge joins a vertex $$a_i$$ for some $$i \in \{1,\dots ,n\}$$ with a vertex $$b_j$$, where $$j$$ is congruent with someone of the numbers $$i+1,\dots ,i+k$$ modulo $$n$$. If each edge of $$C_{n,k}$$ is replaced by $$\lambda$$ edges with the same end vertices as it had, the multicrown $$\lambda C_{n,k}$$ is obtained. It is proved that $$\lambda C_{nk}$$ can be decomposed into copies of the star $$S_{\ell }$$ with $$\ell$$ edges if and only if $$\ell \leq k$$ and $$\lambda _{nk} \equiv 0$$ $$(\text{mod} \ell)$$. This implies also that the $$k$$th power of the cycle $$c_n$$ of length $$n$$ can be decomposed in that way if and only if $$\ell \leq k+1$$ and $$nk\equiv 0$$ $$(\text{mod} \ell)$$.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
crown; star; decomposition