E. Kubicka
ABSTRACT
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 achieve the minimum sum. In fact, we prove that our family gives the smallest trees with this property. Moreover, we show that asymptotically, for each value of k, almost all trees require more than k colors. Finally, we present a linear algorithm for computing the chromatic sum of an arbitrary tree.
- 1.G. Chartrand and L. Lesniak, Graphs & Digraphs 2nd Edition, Wadsworth & Brooks/Cole, Monterey CA 1986. Google Scholar
Digital Library
- 2.M. R. Garey and D. S. Johnson, Computers and Intractability, 1979.Google Scholar
- 3.R. M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computation, Plenum Press, 1972.Google ScholarCross Ref
- 4.A. Schwenk, Almost all trees are cospectral, New Directions in the Theory of Graphs, (F. Harary, ed.) Academic Press, New York, 1973, 155-163.Google Scholar
Index Terms
- An introduction to chromatic sums
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