The search for chromatically unique graphs

The number of vertex-colourings of a simple graphG in not more than colours is a polynomial in. This polynomial, denoted byP(G, ), is called the chromatic polynomial ofG. A graphG is said to be chromatically unique, in short-unique, ifH G for any graphH withP(H, ) = P(G, ). Since the appearance of the first paper on-unique graphs by Chao and Whitehead in 1978, various families of and several results on such graphs have been obtained successively, especially during the last five years. It is the aim of this expository paper to give a survey on most of the works done on-unique graphs. A number of related problems and conjectures are also included.1980 Mathematical Subject Classification. Primary 05C15This work was done while the author was visiting the Department of Mathematics, National University of Singapore.