Graph Coloring Problem

dahl@ee.umn.edu (peter boardhead dahl)Sat, 24 Oct 1992 20:49:55 GMT

From comp.compilers

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 Newsgroups: comp.compilers,comp.theory From: dahl@ee.umn.edu (peter boardhead dahl) Organization: University of Minnesota, Minneapolis, EE dept. Date: Sat, 24 Oct 1992 20:49:55 GMT Keywords: theory, question

We need help with a Graph Coloring Problem!

Facts: A *clique* of size "k" is "k" colorable.

----- -----
| A | ---- | B | This is a clique of size 4.
----- -----
| \ / | => This graph is 4 colorable.
| \/ |
| /\ |
| / \ |
----- -----
| C | ---- | D |
----- -----

Observation: A general Conflict graph is an interconnection of cliques.

The Conflict Graph "G" below is made up of 6 interconnected 3-cliques.
(All the cliques don't have to be the same size, they just are in this
example.) Also the largest clique in this graph is of size "3".

Cliques:
1) A-B-D 4) C-D-F
2) A-C-D 5) D-E-F
3) B-D-E 6) D-F-G

Note: Any clique included in a larger clique is counted as part of the
larger. ex: The 2-clique A-C is counted as part of the 3-clique A-C-D.

Graph: "G"
----- -----
| A | ----- | B |
/ ----- ----- \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
----- ----- -----
| C | ---------- | D | ---------- | E |
----- ----- -----
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ ----- ----- /
| F | ----- | G |
----- -----

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.)

ex: In the graph "G" above, the largest clique is of size 3 and it's
also 3 colorable. ( works in this case, how about others? )
--
Peter Bergner (bergner@ee.umn.edu)
Peter Dahl (dahl@ee.umn.edu)
--

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