How to color a map, so that one color covers the maximum area. Given a graph $G$ where every node resembles a number. How would one color this graph, so that no two connected nodes have the same color and that one color has the maximum of area? Area is defined by the number inside the node.

For this example problem I found a solution of $22$. This is achieved by coloring the number 8, the lower 5, the highest 3, the upper right 4 and the remaining 2. I'd like to know if there is a good method for finding this number without using trial and error.
EDIT: You can use as many colors as you like, so the four color theorem does not apply here.

I believe I worked out a solution to the example problem and a decent method to get there. Copy of the problem:

First I decided that it wasn't needed to label all of them with the same color. There's is only one color we're actually interested in. So we can label every node with R (Red) or O (Other).
For the 8 we have two options: R or O. If we color it in with O it must mean that we would have to color the 4, the 2 or the 3 with R. Because if we were to cover them all with O, we would just lose the 8 points for no reason. So what is the maximum amount of points we can get by coloring the 4, 2 and 3? Seven of course (color the 4 and the 3). But 7 is less then 8 so you'd have to color 8. There are basically two options:
8: R, 4: O, 2: O, 3: O --> this results in a score of 8.
8: O, 4: R, 2: O, 3: R --> results in a score of 7 and you also make 3 other nodes (the 5, 2 and 3) O, so they can't contribute any points.
So I color the 8 Red. The 4, 2 and 3 must be any Other color.
No I can lift the 8, 4, 2 and 3 out of the graph. They have been colored in so they don't matter anymore. Now you could simplify the problem to this:
simplified problem
You could go through the problem one more time but instead of looking at the 8, you can look at the lower 5. Again there are two options:
5: R, 2: O, 3: O, 2: O --> results in a score of 5.
5: O, 2: R, 3: O, 2: R --> results in a score of 4 and covers up 4 other nodes with O.
So I color the 5 red and the 2, 3 and 2 must be any Other color.
rince and repeat:
More Simplified Graph
Take a look at the two's at the left.
Two options:
Lowest 2: R, higher 2: O --> results in a score of 2.
Lowest 2: 0, higher 2: R --> results in a score of 2 and you cover up the 4.
Take the Lowest 2. Simplify.
Last Simplified Version
From here I can see that, to take the maximum amount, I now need to take the 3 and the 4 to get the maximum in this graph. And at this point we're done. We have a total of: 8 + 5 + 2 + 4 + 3 = 22.
NOTE: This sudoku-ing of the graph doesn't always work (just like how not all beginning positions of a sudoko are solvable). I tried for example with the map of the United States: I got stuck. The 'algorithm' I used is still extremely cumbersome and takes a lot of time.
