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?