Tight bounds on the chromatic sum of a connected graph

The chromatic sum of a graph is introduced in the dissertation of Ewa Kubicka. It is the smallest possible total among all proper colorings of G using natural numbers. In this article we determine tight bounds on the chromatic sum of a connected graph with e edges, 1. THE CHROMATIC SUM A proper coloring of the vertices of the graph G must assign different colors to adjacent vertices. The chromatic number x(G) is just the smallest number of colors in any proper coloring of G. The chromatic number is well known and much studied. The reader may seek background in any graph theory text, for example, Chartrand and Lesniak [l]. The chromatic sum Z(G) is a recent variation introduced in the dissertation of Ewa Kubicka [2]. It is defined as the smallest possible total over all vertices that can occur among all proper colorings of G using natural numbers for the colors (as is customary). It is tempting to suspect that we will attain the minimum sum by first selecting a coloring that achieves the chromatic number and then arranging the color classes so that the largest is color 1, the next largest is color 2, and so on. But it is shown in [2] that even among trees (whose chromatic number is clearly 2) the chromatic sum sometimes requires the use of more than 2 colors; in fact, it is shown that for every positive integer k nearly all trees require the use of at least k colors to attain the chromatic sum. Thus, in the long run, we cannot expect a coloring Journal of Graph Theory, Vol. 13, No 3. 353-357 (1989)