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# 3SAT23COL - R | | | L1 L2 L3 | | | | | | X-Y | | | | Z | |...

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R / | \ / | \ / | \ L 1 L 2 L 3 | | | | | | X-------------------------Y | \ / | \ / | \ / | \ / | Z | | | | | D----------------------------------------E \ / \ / \ / \ / 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. If a gadget is 3-colorable, then at least one of L 1 , L 2 , or L 3 must have the same color as C, that is colored 1.

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