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$$i$$-perfect $$m$$-cycle systems, $$m\leq 19$$. (English) Zbl 0883.05110
An $$m$$-cycle decomposition of a graph $$G$$ is a pair $$(V,C)$$, where $$V$$ is the vertex set of $$G$$, and $$C$$ is a collection of edge-disjoint $$m$$-cycles in $$G$$ which cover its edges. In the case $$G$$ is the complete graph, $$K_v$$, $$(V(G),C)$$ is called an $$m$$-cycle system of order $$v$$. If $$c$$ is a cycle of length $$m$$, then let $$c(i)$$ denote the graph formed from $$c$$ by joining all vertices in $$c$$ at distance $$i$$. If $$(V,C)$$ is an $$m$$-cycle decomposition of $$G$$ such that $$(V, \{c(i) \mid c\in C\})$$ is also a cycle decomposition of $$G$$, then we call $$(V,C)$$ an $$i$$-perfect $$m$$-cycle decomposition of $$G$$. For all $$m\leq 19$$ and each meaningful value of $$i$$ ($$2\leq i \leq \lfloor m/2\rfloor$$), the spectrum problem for $$i$$-perfect $$m$$-cycle systems is examined. The authors describe several constructions for $$i$$-perfect $$m$$-cycle systems and they construct several new $$i$$-perfect $$m$$-cycle systems for $$m\leq 19$$. The paper contains a table summarizing all of the new and known results for $$i$$-perfect $$m$$-cycle systems for $$5\leq m\leq 19$$.

MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05B30 Other designs, configurations 05C38 Paths and cycles
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References:
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