The chromaticity of wheels with a missing spoke II

In the previous paper, it was shown that the graph U. ÷ 1 obtained from the wheel W n ÷ 1 by deleting a spoke is uniquely determined by its chromatic polynomial if n >i 3 is odd. In this paper, we show that the result is also true for even n >~ 4 except when n = 6 in which case, the graph W given in the paper is the only graph having the same chromatic polynomial as that of U 7. The relevant tool is the notion of nearly uniquely colorable graph. As in [2], only finite undirected graphs without loops or multiple edges will be considered. A graph G is chromatically unique if it is uniquely determined by its chromatic polynomial P(G;2). The wheel IV.+ 1 is obtained by taking the join of a single vertex and the cycle Cn on n vertices. The graph U.+ ~ is obtained from Wn÷l by deleting a spoke which is an edge joining the single vertex to a vertex on Cn. In [2], it is shown that Un + 1 is chromatically unique if n >~ 3 is odd. In this paper we shall show that the result extends to all n ~> 3 except for n = 6 in which case, the graph W of Fig. 1 is the only graph having the same chromatic polynomial as that of UT. The relevant tool is the notion of nearly uniquely colorable graph. A graph is nearly uniquely s-colorable if it has chromatic number s and there are precisely two ways of partitioning its vertex set into s independent subsets, up to permutation of these independent subsets. In terms of chromatic polynomial, if a graph G is nearly uniquely s-colorable, then P(G;s) = 2(s!). Notice that if n/> 4 is even, then U~÷~ is nearly uniquely 3-colorable. Another example of a nearly uniquely 3-colorable graph is the graph W of Fig. 1. Suppose G is nearly uniquely s-colorable. Let Vo . . . . . Vsx be the color classes of an s-coloring of G. Then for any 0 ~< i ~< s 1, we may write the color class V~ as V~ = Zi w Xi where Zi is the set of those vertices always sharing the same color in any s-coloring of G; X~ has the similar property except that in a different s-coloring, all vertices in X~ may get possibly a color different from those in Zi. 0012-365X/96/$15.00 © 1996--Elsevier Science B.V. All rights reserved SSDI 0012-365X(94)00248-7 306 G.L. Chia / Discrete Mathematics 148 (1996) 305-310