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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

05C15 Coloring of graphs and hypergraphs
Full Text: DOI
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