![]() | THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5. |
Here's the Venn graphs of Venn's general construction for n=3,4,5. The red edges indicate the Hamilton cycle that is used in extending to the next higher value of n.
How, in general, do you go from the dual (Venn) graph G of a Venn diagram, together with a Hamilton cycle H in G, and get a new planar dual (Venn) graph G' of a Venn diagram of the next higher order? We now explain this process. Note that H is a simple closed curve with an interior and an exterior.
We illustrate the discussion on the expansion of n=4 to n = 5 as shown above. Color the edges of H red, edges on the interior blue, and edges on the exterior black.
Note that the red edges in G' give the prism of the Hamilton cycle in G.
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![]() | THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5. |