Dean G. Hoffman

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In this paper we use Tutte's f-factor theorem and the method of amalgamations to find necessary and sufficient conditions for the existence of a k-factor in the complete multipartite graph K(p(1 ) ..... p(n)), conditions that are reminiscent of the Erd6s-Gallai conditions for the existence of simple graphs with a given degree sequence. We then use this(More)
Philip Hall’s famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or listmulticolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural(More)
A 4-cycle decomposition of a complete multipartite graph is said to be gregarious if each 4-cycle in the decomposition has its vertices in four different partite sets. Here we exhibit gregarious 4-cycle decompositions of the complete equipartite graph Kn(m) (with n 4 parts of size m) whenever a 4-cycle decomposition (gregarious or not) is possible, and also(More)