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R
/

\
/

\
/

\
L
1
L
2
L
3






XY

\
/

\
/

\
/

\
/

Z





DE
\
/
\
/
\
/
\
/
C



R
Sorry, this is intended to be the same picture as in the document I
sent last evening. I have given labels to every vertex. The top and
bottom vertex are actually the same vertex. The L
i
vertices are U and
"U bar" vertices as described in the document. This is just the"gadget"
for one clause.
For each clause there is a similar gadget, and all have
vertices R and C in common. They also share L
i
vertices when the
clauses have a common variable.
In any proper coloring of this gadget (3 colors or not) we may
assume R has a color and specify it to be 3. We know C will receive a
color  it can not be 3, but it can be any other color we want. So we will
arbitrarily choose it to be 1. We also know each
L
i
will have a color and
it will also not be 3, but we can not say what or how many at this point.
Lemma 1.
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This note was uploaded on 07/14/2011 for the course COT 4610 taught by Professor Dutton during the Fall '10 term at University of Central Florida.
 Fall '10
 Dutton

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