User:Stixme/Sandbox

Every Simple Planar Graph is at least 6-colorable
For any simple planar graph $G_n$, where $n$ is a number of vertices, the following is true:
 * $\chi \left(G_k \right) \leq 6$, where $\chi$ is a chromatic number of a graph.

Without loss of generality we can assume that the graph $G_n$ is connected.

Proof
We will use induction on the number of vertices to prove the theorem:

For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * $G_n$ can be assigned a proper vertex $k$-coloring such that $k \le 6$.

Base of Induction
$P \left({r}\right)$ is trivially true for $1 \le r \le 6$, as there are no more than $6$ vertices to be colored.

Induction Hypothesis
Consider a planar graph $G_{r+1}$.

By the Minimum Degree Bound:


 * $\exists v, \: v\in V(G_{r+1})$, such that $δ(v) ≤ 5$.

Consider a graph $G'=G_{r+1}-v$.

By the induction hypothesis we can 6-color $G'$.

Thus, $G_{r+1}$ has each vertex colored without conflicts except for the $v$.

Since there are at most 5 colors adjacent, we have at least one color left.

Use an available color for $v$. We have then 6-colored the graph.