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POColoring specific type of the planar graph
 

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  1. POColoring specific type of the planar graph
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    <p>I've got a specific type of the planar graph and I found it interesting to search for an algorithm which will color its vertices <a href="http://en.wikipedia.org/wiki/Graph_coloring" rel="nofollow">legally</a>. About this type of graph, it's very easy and cool:</p> <blockquote> <p>Consider any tree <strong>T</strong> with <strong>n>2</strong> vertices and <strong>k</strong> leaves. Let's denote <strong>G(T)</strong> as a graph constructed from <strong>T</strong> by connecting its leaves into <strong>k</strong>-cycle in such way that <strong>G(T)</strong> is planar.</p> </blockquote> <p>And the problem I came up with is to color <strong>G(T)</strong> with <strong>3</strong> colors. Clearly, <strong>G(T)</strong> as a planar graph, is <strong>4</strong>-colorable, but I think (don't have a proof) that it is almost always <strong>3</strong>-colorable due to its simplicity. Almost always means that only if <strong>T</strong> is a <a href="http://en.wikipedia.org/wiki/Star_%28graph_theory%29" rel="nofollow">star</a> and only with odd number of leaves, then <strong>G(T)</strong> is <strong>4</strong>-colorable.</p> <p>I am looking for some algorithm, or maybe proof of my assumptions which could be easily transformed into an algorithm. I would be very very grateful for any help, hints. </p> <p>In case I wasn't clear enough I'll give an example:</p> <p>Let T be a tree with edges E(T) = { {1,2}, {2,3}, {2,4}, {4,5} } and then E(G(T)) = sum of the sets: E(T) and { {1,5}, {5,3}, {3,1} }, since we are connecting leaves 1,5,3 into a cycle. </p>
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    1. COAnd what about vertex #4 ?
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      1. USEgor Skriptunoff
    2. CO@EgorSkriptunoff, you are right, it was a typo, sorry. I just fixed this.
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    3. CO@xan Did you hear about the Grötzsch theorem for triangle-free planar graphs? http://en.wikipedia.org/wiki/Gr%C3%B6tzsch's_theorem
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      1. USDaniel Martín
 

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