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On the chromatic uniqueness of certain bipartite graphs. (English) Zbl 0752.05030
A graph $$G$$ is chromatically unique if any other graph with the same chromatic polynomial is isomorphic to $$G$$. For example, it is known that the complete bipartite graph $$K(n,m)$$ is chromatically unique provided $$n,m\geq 2$$, as is the complete graph minus a single edge $$K^{-1}(n,m)$$ provided $$n,m\geq 3$$.
In this paper the author extends the investigation into the chromatic uniqueness of complete bipartite graphs minus 2, 3, or 4 edges. Let $$K^{-r}(n,m)$$ denote the class of complete bipartite graphs $$K(n,m)$$ with $$r$$ edges removed. There are 3 graphs in $$K^{-r}(n,m)$$ when $$r=2$$, 6 graphs when $$r=3$$, and 16 graphs for $$r=4$$. The author derives a sufficient condition for a graph in $$K^{-2}(n,m)$$ to be chromatically unique. In particular, each such graph is chromatically unique when $$| n-m|\leq 3$$. Moreover, graphs with $$| n-m|=d$$ are chromatically unique provided that $$n$$ is sufficiently large (an explicit polynomial bound is given). In the class $$K^{-3}(m,m)$$ any member is chromatically unique provided $$| n-m|\leq 1$$ (with a few small exceptions). In $$K^{-4}(n,m)$$ any graph is chromatically unique provided $$n=m\geq 4$$. Also, each graph in $$K^{-4}(n,n+1)$$ is chromatically unique for $$n\geq 5$$. A few other results of a similar nature are presented.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
chromatic uniqueness; chromatic polynomial
Full Text:
##### References:
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