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On the chromatic uniqueness of bipartite graphs. (English) Zbl 0594.05034
From the authors’ abstract: ”We prove the chromatic uniqueness of the following infinite families of bipartite graphs: $$\bar K_ r\cup K_{m,m+k}$$, with $$M\geq 2$$ and $$0\leq k\leq \max (5,\sqrt{2m})$$. We have also proved that $$K^-_{m,m+k},\bar K_ r\cup K^-_{m,m+k}$$ are chromatically unique for $$k=0,1$$ and $$m\geq 3$$, where $$K^-_{m,n}$$ denotes the graph obtained from $$K_{m,n}$$ by deleting one edge. As a particular case we prove a conjecture made by C. Y. Chao in 1978 that $$K_{m,m+k}$$ is chromatically unique for $$0\leq k\leq 2$$ and $$m\geq 2.''$$
Also the following conjecture is proposed: $$K_{m,n}$$ is chromatically unique for $$n\geq m>1$$. The method of proof is by evaluating the number of 4-cycles in some bipartite graphs. The reviewer showed recently that $$K_{m,m+k}$$ is chromatically unique for $$m\geq 2$$ and $$0\leq k<2\sqrt{m+1}$$ by considering an extremal property of 3-colourings of bipartite complete graphs [On 3-colourings of bipartite p-threshold graphs, J. Graph Theory (to appear)].
Reviewer: I.Tomescu

##### MSC:
 05C15 Coloring of graphs and hypergraphs
Full Text:
##### References:
 [1] Chao, C.Y; Whitehead, E.G, Chromatically unique graphs, Discrete math., 27, 171-177, (1979) · Zbl 0411.05035 [2] Eisenberg, B, On the coefficients of the chromatic polynomial of a graph, () [3] Farrel, E.J, On chromatic coefficients, Discrete math., 29, 257-264, (1980) · Zbl 0443.05041 [4] Frucht, R.W; Guidici, R.E, Some chromatically unique graphs with seven points, Ars combin., 16-A, 161-172, (1983) · Zbl 0536.05026 [5] Giudici, R.E, Some new families of chromatically unique graphs, (), 147-158 [6] Harary, F, Graph theory, (1969), Addison-Wesley Reading, MA · Zbl 0797.05064 [7] Read, R.C, An introduction to chromatic polynomials, J. combin. theory, 4, 52-71, (1962) · Zbl 0173.26203 [8] Whitehead, E.G; Zhao, L.C, Cutpoints and the chromatic polynomial, J. graph theory, 4, 371-377, (1984) · Zbl 0551.05041
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