[r-t] Smith's Theorem

[Richard Smith]

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