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Thread: 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 $\displaystyle P(n)$ be the proposition 'every tree with n vertices is bipartite.' The $\displaystyle P(1)$ is true, which is trivial.

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

    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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