Date: Jun 18, 2013 11:36 PM
Author: quasi
Subject: Re: A New Graph Coloring Conjecture.

Zeit Geist wrote:
>quasi wrote:
>> bill wrote:
>> >
>> >Conjecture: A graph is four-colorable, if it does not contain
>> >a complete K5 sub graph.

>>
>> I think the following works as a counterexample ...
>>
>> Let G = (V,E) where
>>
>> ...
>>
>> Therefore, if I haven't made an error, G serves as a
>> counterexample to the stated claim.

>
>You are correct.
>
>So, I am wondering,
>
>What it is that the OP, bill, is investigating?

I guess he's trying to find a simple test to determine whether
or not a given graph G is 4-colorable.

I'll ask the question this way:

Does there exist a finite set of non-4-colorable graphs such
that, if a given graph G has no subgraphs isomorphic to one of
those graphs, then G is 4-colorable?

Another way to ask the question is this:

Consider the set of non-4-colorable graphs such that every
proper subgraph is 4-colorable. Are there only finitely many
such graphs, up to isomorphism?

I think it's clear that the two questions above are equivalent,
and I'm fairly sure that the answer to those question is no.

quasi