The Chromaticity of Certain Complete Multipartite Graphs

In this paper, we first establish a useful inequality on the minimum real roots of the adjoint polynomials of the complete graphs. By using it, we investigate the chromatic uniqueness of certain complete multipartite graphs. An unsolved problem (i.e., Problem 11), posed by Koh and Teo in Graph and Combin. 6(1990) 259–285, is completely solved by giving it a positive answer. Moreover, many existing results on the chromatic uniqueness of complete multipartite graphs are generalized.Acknowledgments. The authors are greatly indebted to the referees for their valuable comments and suggestions, which are very helpful for improving the presentation of the paper. The work is supported by National Science Foundation of China and the Science Foundation of the State Education Ministry of China.Final version received: January 9, 2004AMS Subject Classification (2000): 05C15, 05C60