Sat, 31 Oct 1992 01:02:17 GMT

Related articles |
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[2 earlier articles] |

Re: Graph Coloring Problem jrbd@craycos.com (1992-10-27) |

Re: Graph Coloring Problem pat%frumious.uucp@uunet.ca (1992-10-28) |

Re: Graph Coloring Problem Richter@lrz.lrz-muenchen.dbp.de (1992-10-28) |

Re: Graph Coloring Problem cliffc@rice.edu (1992-10-28) |

Re: Graph Coloring Problem moss@cs.umass.edu (1992-10-28) |

Re: Graph Coloring Problem preston@cs.rice.edu (1992-10-30) |

Re: Graph Coloring Problem sgall+@CS.CMU.EDU (1992-10-31) |

Re: Graph Coloring Problem kuzemcha@tartan.com (1992-11-06) |

Newsgroups: | comp.compilers,comp.theory |

From: | sgall+@CS.CMU.EDU (Jiri Sgall) |

Organization: | Carnegie Mellon University |

Date: | Sat, 31 Oct 1992 01:02:17 GMT |

References: | 92-10-093 92-10-107 |

Keywords: | theory |

|> >QUESTION: Given a Conflict graph "G" in which the largest clique

|> > in the graph is of size "k", is the graph "k" colorable?

|>

|> Well, it took about 100 years to prove this for the specific case k=4, so

|> don't expect the general proof to be very easy (hint: a planar graph can't

|> have a clique of size 5 :-).

Obviously this does not prove anything relevant. Being planar is much

stronger property than not having a clique of size 5.

In fact, Erdos in 1959 proved that for any k and l there is a graph that

is not k colorable and does not contain a circle shorter than l. Any

clique contains triangle, i.e. 3-cycle, hence it is trivial consequence

that James' claim is not true for any k.

For the proof see e.g. Alon, Spencer, The Probabilistic Methods, chapter

3. I beleive that there are constructive examples of these graphs, but

the general construction is much more complicated that the probabilistic

proof (as usual).

-- Jiri Sgall

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