Guide

Definitions, theory, and references.

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The table at left lists, for 2<=n<=11, all possible partitions of the integer n into parts of size larger than one. For example, the partition of 8 into 3+3+2 is encoded as [3,3,2]. For each such partition [k₁,k₂, ...,k_s], we ask "Is there a one-factorisation of the complete bipartite graph K_n,n having the property that the union of any two distinct one-factors is isomorphic to a 2-regular graph having components of size 2k₁, 2k₂,..., 2k₁?" The table indicates "Yes" if an example is known, "No" if it is known that no example exists, and "?" in the remaining cases. If one clicks on any link in the table, an example, construction, or non-existence argument appears in the frame above.