@inproceedings{Kubicka1989AnIT,
title={An Introduction to Chromatic Sums},
author={Ewa Kubicka and Allen J. Schwenk},
booktitle={ACM Conference on Computer Science},
year={1989}
}

We introduce the new concept of the chromatic sum of a graph G, the smallest possible total among all proper colorings of G using natural numbers. We show that computing the chromatic sum for arbitrary graphs is an NP-complete problem. Indeed, a polynomial algorithm for the chromatic sum would be easily modified to compute the chromatic number. Even for trees the chromatic sum is far from trivial. We construct a family of trees to demonstrate that for each k, some trees need k colors to achieveâ€¦Â CONTINUE READING