# Re: Graph Coloring Problem

## preston@cs.rice.edu (Preston Briggs)Fri, 30 Oct 1992 01:16:06 GMT

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 Newsgroups: comp.compilers,comp.theory From: preston@cs.rice.edu (Preston Briggs) Organization: Rice University, Houston Date: Fri, 30 Oct 1992 01:16:06 GMT References: 92-10-093 Keywords: theory

dahl@ee.umn.edu (peter boardhead dahl) writes:

>QUESTION: Given a Conflict graph "G" in which the largest clique
> in the graph is of size "k", is the graph "k" colorable?
> (It seems to be true.)

Hi Peter,

I think not. Here's a counter-example.
Look at a string of 4-cliques. I'll draw the basic clique like this

0 - 2
| X |
1 - 3

that is, four vertices: 0, 1, 2, 3
and 6 edges: 0-1, 0-2, 0-3, 1-3, 1-2, 2-3

The string would look like

0 - 2 - 4 - 6 - 8
| X | X | X | X |
1 - 3 - 5 - 7 - 9

Assigning colors (say, a, b, c, and d), we get

a - c - a - c - a
| X | X | X | X |
b - d - b - d - b

which is fine. Now suppose we connect the ends of our string,
creating a new clique, with the edges

0-8, 1-8, 0-9, 1-9

The resulting graph can't be colored in only four colors;
but the largest clique still has only 4 vertices.

Here's the smallest counterexample.

Consider a pentagon.
The largest clique is 2, but it requires 3 colors.

1 - 2
| |
2 1
\ /
3

Of course, it's easy to make larger cases.
All simple cycles with an odd number of vertices require 3 colors,
and in all cases with more than 3 vertices, the largest clique is 2.

Preston Briggs
--

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