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On smallest regular graphs with a given isopart. (English) Zbl 0607.05039
A G-decomposition of a graph H is a family of subgraphs \(H_ 1,H_ 2,...,H_ n\) of H, whose edge-sets partition E(H), and such that each \(H_ i\) is isomorphic to G. G-decompositions have been studied by Bermond, Rosa, Schönheim, Harary, Wilson, and many others. Here the author asks for three extremal parameters related to G-decompositions: the least p (respectively r, respectively f) such that there exists a connected regular graph H which admits a G-decomposition and has p vertices (respectively degree r, respectively the G-decomposition has f subgraphs). After a fairly straightforward calculation of the parameters p, r, and f for complete graphs, cycles, and stars, the author studies the parameter r for an arbitrary tree T. If the maximum degree \(\Delta\) (T) is even, then \(r=\Delta (T)\). Otherwise r could be \(\Delta\) (T) or \(\Delta (T)+1\); when \(\Delta (T)=3\) the author characterizes those trees with \(r=3\).
Reviewer: P.Hell

MSC:
05C35 Extremal problems in graph theory
05C05 Trees
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