Consider the following coloring algorithm for planar graphs:
- Create a stack $S$ with the vertices of $G$ as follows:
(a) If $v$ is has the lowest degree vertex in $G$, add $v$ to $S$ and change to $G$ for $G-{v}$.
The first vertex of $S$, color it with color 1
The next vertex of $S$ color it from the minimum integer that has not been used on any of its neighbors.
Repeat 3. until all vertices are colored.
How would you improve the algorithm in such a way that the resulting coloration always use at most 5 colors?
I've been trying to improve this algorithm as it is required, but since this algoritm is kind of Greddy Algorithm "brother" we can use at most 6 colors, since every vertex in a planar graph has at most $d(v) ≤ 5$, How can I improve this? I think the part we can improve this is between step 3 and 4