×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.