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Math Help - 2 colourable tree

  1. #1
    Senior Member
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    2 colourable tree

    Prove that all trees are 2-colourable, from this are all trees there planar or not? Justify your answer.

    I am just stuck on how to start this question, how do I prove all trees are 2-colourable...

    Many thanks!
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  2. #2
    Member Traveller's Avatar
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    Use induction. Use the facts that if we remove a leaf from a tree it still remains a tree and a leaf is adjacent to only one vertex.
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  3. #3
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    Quote Originally Posted by usagi_killer View Post
    Prove that all trees are 2-colourable, from this are all trees there planar or not? Justify your answer.

    I am just stuck on how to start this question, how do I prove all trees are 2-colourable...

    Many thanks!
    First, a graph is bipartite if it can be colored with two colors.
    So lets prove by induction that a tree is bipartite.

    Let P(n) be the proposition 'every tree with n vertices is bipartite.' The P(1) is true, which is trivial.

    Suppose that m \ge 1 and that P(m) is true. Let T be a tree with m+1 vertices. Then T contains a leaf x. Let y be be the vertex adjacent to x and let T^' be the tree formed by removing x from T. Then T^' has a 2-coloring, and we can extend this to T by coloring x a different color to y.

    Thus every tree is 2-colorable.

    As to the question about if all trees are planar, I wouldn't know what to do. What are your thoughts on this?
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