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Topic: A New Graph Coloring Conjecture.
Replies: 42   Last Post: Jul 2, 2013 4:08 AM

 Messages: [ Previous | Next ]
 b92057@yahoo.com Posts: 1,187 Registered: 4/18/05
Re: A New Graph Coloring Conjecture.
Posted: Jun 20, 2013 12:10 AM

On Tuesday, June 18, 2013 8:21:53 PM UTC-7, quasi wrote:
> 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.
> javascript:;
>
>
> quasi

Okay.

Finite planar graphs are 4-C because they cannot
contain subgraphs isomorphic to K5!

My thinking is thus. In any K4 subgraph one vertex is
unavailable. Therefore, there is no way to force any
set of 4 vertices to have four different colors.
Tangled chains are not due to the congiguration of the
graph but result from a fortuitous coloring. Plus,
if two chains are tangled; there will be two that are
not tangled.

If there are no forceable sets of four vertices, the graph must be 4 colorable.

bill,

Date Subject Author
6/18/13 b92057@yahoo.com
6/18/13 Tucsondrew@me.com
6/18/13 Tucsondrew@me.com
6/18/13 quasi
6/18/13 Tucsondrew@me.com
6/18/13 quasi
6/19/13 trj
6/20/13 b92057@yahoo.com
6/20/13 quasi
6/20/13 b92057@yahoo.com
6/22/13 b92057@yahoo.com
6/23/13 quasi
6/25/13 b92057@yahoo.com
6/25/13 quasi
6/25/13 b92057@yahoo.com
6/26/13 quasi
6/26/13 Brian Q. Hutchings
6/27/13 b92057@yahoo.com
6/27/13 quasi
6/27/13 b92057@yahoo.com
6/27/13 quasi
6/27/13 quasi
6/28/13 b92057@yahoo.com
6/28/13 quasi
6/28/13 b92057@yahoo.com
6/29/13 quasi
7/1/13 b92057@yahoo.com
7/2/13 quasi
6/19/13 b92057@yahoo.com
6/19/13 quasi
6/20/13 b92057@yahoo.com
6/20/13 Tucsondrew@me.com
6/21/13 quasi
6/21/13 quasi
6/22/13 b92057@yahoo.com
6/23/13 quasi
6/19/13 b92057@yahoo.com
6/20/13 quasi
6/20/13 quasi
6/20/13 Butch Malahide
6/20/13 Butch Malahide
6/19/13 b92057@yahoo.com
6/20/13 b92057@yahoo.com