Creating a problem that has a unique solution (like the one above) is less obvious, or isn’t it ? ]]>

Then there are only two 4-4 pairs, and one of them must be correct.

You should follow both “4-4” paths consequently.

If e.g. the 4-4 on the left is right, then also the 2-4 pair in the upper left corner should be right, etcetera.

If there are any forced doubles left, then the followed path is wrong.

By means of making trees and elimination the answer should be found, but I am not going to write down all these numbers in tenfold… ]]>

Two observations:

1) If we seek independent sets with the maximum number of colors represented, then we never need to consider an independent set with two vertices of the same color: we can simply throw out duplicates.

2) Consequently, the problem of finding the independent set with maximum number of colors reduces to the problem of finding an independent set of maximum size on the following modification of your graph: after taking the line graph, draw extra edges connecting any pair of vertices of the same color. (An independent set on this graph is the same as an independent set on the line graph that contains no two vertices of the same color.)