Objects, Invariants and Properties for Graph Theory (GT) automated conjecturing: in particular with the Sage program CONJECTURING: http://nvcleemp.github.io/conjecturing/
The chromatic_sum is the smallest sum of vertex labels, when the vertices are labeled with positive integers and adjacent labels are labeled with different integers ("colors").
This is intractable, and at least as hard as chromatic_number. A naive algorithm is: find the chromatic number k, then try all possible labels of the vertices with 1,2..,k, check if its a proper coloring, compute the sum, and store the current minimum.
That naive algorithm is in fact not slow enough 😞. According to the wiki, achieving the unique chromatic sum may require more labels than the chromatic number. So, we would have to check even more than 1,2,..,k.
This was discussed at SIAM DM'18.
The chromatic_sum is the smallest sum of vertex labels, when the vertices are labeled with positive integers and adjacent labels are labeled with different integers ("colors").
See: https://en.wikipedia.org/wiki/Sum_coloring
This is intractable, and at least as hard as chromatic_number. A naive algorithm is: find the chromatic number k, then try all possible labels of the vertices with 1,2..,k, check if its a proper coloring, compute the sum, and store the current minimum.