2 colourable tree

• Oct 12th 2010, 03:27 AM
usagi_killer
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!
• Oct 12th 2010, 04:04 AM
Traveller
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.
• Oct 12th 2010, 06:18 PM
Nguyen
Quote:

Originally Posted by usagi_killer
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?