[r-t] Smith's Theorem
Richard Smith
richard at ex-parrot.com
Sun Sep 24 12:24:12 UTC 2017
This "theorem" turns out to be a simple corollary of a
rather more powerful result that's been discussed and proved
several times on this list, for example in a post by me on
18 May 2011 in regard to Double Cambridge Cyclic Bob Minor,
though I don't claim to have discovered it. That result is
that, in scenarios when you need to include precisely half
the available leads and they are to be joined only by
lead-end calls, the falseness graph (the Cayley graph
generated by the false lead heads in the set of available
leads) must be bipartite. This is a necessary condition for
an extent to exist, and a necessary and sufficient condition
for there to be a set of mutually true leads covering the
extent.
The false lead head generated from any two rows in a lead of
method is conjugate to the transposition relating those two
rows, so if those rows are related by a transposition of odd
order, the false lead head also has odd order. If the
treble was in the same place in the two rows, the false ead
head is a fixed-treble one, so is a generator of the
falseness Cayley graph. It is trivially obvious that a
Cayley graph with a generator of odd order cannot be
bipartite.
But while Robin's result about transpositons of odd order is
a necessary condition for an extent to exist, it is not a
sufficient condition, even for the existence of a set of
mutually true leads. In this sense, the biparticity
requirement is better, especially as biparticity is an easy
property to determine computationally.
RAS
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