Monday, November 30, 2009
lecture 20
Graph coloring is perhaps the most famous graph problem, thanks to the map coloring problem and the 4-color theorem. We first discuss the basic definition and show that a graph is 2-colorable if and only if it is bipartite. Then we briefly mention some applications of graph coloring, and coloring of an interval graphs. Finally we study the map coloring problem and prove that every planar graph is 6-colorable (and then quickly sketch the proof for 5-coloring). The proof is based on a sequence of simple statements, each of which is not difficult to understand but putting it together is not easy. This is the final lecture of this course.
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I would like to ask that in the proof of six colours thm, why the claim is:
ReplyDelete"Every simple planar graph has a vertex of degree at most 5", but not
" All vertexs of every simple planar graph have degree at most 5"?
I believe you are able to tell the difference between the existential and universal quantifiers (exists vs. for all)
ReplyDeleteI know the difference between them...
ReplyDeleteI am wonder why we should not use "for all" this time.
We want to confirm every vertice has degree <=5, then we just say there exist?
regard
Sorry, I think I have misunderstood the proof
ReplyDeleteThe proof just wants to show there exists such vertex and hence in MI, we can isolate that vertex first. Am I right?
Thank You
you're right
ReplyDelete