# 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 \$\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?