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We give a self‐contained proof that for all positive integers r and all , there is an integer such that for all any regular multigraph of order 2n with multiplicity at most r and degree at least is 1‐factorizable. This generalizes results of Perković and Reed (Discrete Math 165/166 (1997), 567–578) and Plantholt and Tipnis (J London Math Soc 44 (1991), 393–400).
A gerechte framework is a partition of an n × n array into n regions of n cells each. A realization of a gerechte framework is a latin square of order n with the property that when its cells are partitioned by the framework, each region contains exactly one copy of each symbol. A gerechte design is a gerechte framework together with a realization.
We investigate gerechte frameworks where each region...
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