The search for chromatically unique graphs - II

The number of vertex-colourings of a simple graphG in not more thanλ colours is a polynomial inλ. This polynomial, denoted byP(G, λ), is called the chromatic polynomial ofG. A graphG is said to be chromatically unique, in shortχ-unique, ifH ≅ G for any graphH withP(H, λ) = P(G, λ). Since the appearance of the first paper onχ-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 onχ-unique graphs. A number of related problems and conjectures are also included.

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