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Chromatic classes of 2-connected \((n,n+3)\)-graphs with at least two triangles. (English) Zbl 0796.05033
Two graphs \(G\) and \(H\) are said to be chromatically equivalent if their chromatic polynomials are the same. A graph \(G\) is said to be chromatically unique if every graph chromatically equivalent to \(G\) is isomorphic to \(G\). Let \(C\) denote the class of all 2-connected graphs of order \(n\) and size \(n+3\) having at least two 3-cycles. In this paper all equivalence classes in \(C\) under the equivalence relation of chromatic equivalence are determined; the structure of the graphs in each class is characterized; new families of chromatically equivalent and chromatically unique graphs are produced.

05C15 Coloring of graphs and hypergraphs
05C38 Paths and cycles
Full Text: DOI
[1] Chao, C. Y.; Whitehead, E. G., Chromatically unique graphs, Discrete Math., 27, 171-177, (1979) · Zbl 0411.05035
[2] Chao, C. Y.; Zhao, L. C., Chromatic polynomials of a family of graphs, Ars Combin., 15, 111-129, (1983) · Zbl 0532.05027
[3] Farell, E. J., On chromatic coefficients, Discrete Math., 29, 257-264, (1980) · Zbl 0443.05041
[4] Koh, K. M.; Goh, B. H., Two classes of chromatically unique graphs, Discrete Math., 82, 13-24, (1990) · Zbl 0697.05027
[5] Li, W. M., Almost every K_4-homeomorph is chromatically unique, Ars Combin., 23, 13-36, (1987) · Zbl 0644.05020
[6] Loerinc, B., Chromatic uniqueness of the generalized θ-graphs, Discrete Math., 23, 313-316, (1978) · Zbl 0389.05034
[7] Read, R. C., An introduction to chromatic polynomials, J. Combin. Theory, 4, 52-71, (1968) · Zbl 0173.26203
[8] Teo, K. L.; Koh, K. M., Chromatic classes of certain 2-connected (n, n+2)-graphs, Ars Combin., 32, 65-76, (1991) · Zbl 0760.05044
[9] Whitehead, E. G.; Zhao, L. C., Cutpoints and the chromatic polynomial, J. Graph Theory, 8, 371-377, (1984) · Zbl 0551.05041
[10] Whitehead, E. G.; Zhao, L. C., Chromatic uniqueness and equivalence of K_4 homeomorphs, J. Graph Theory, 8, 355-364, (1984) · Zbl 0555.05035
[11] Whitney, H., A logical expansion in mathematics, Bull. Amer. Math. Soc., 38, 572-579, (1932) · JFM 58.0605.08
[12] Whitney, H., The colouring of graphs, Ann. Math., 33, 688-718, (1932) · JFM 58.0606.01
[13] Woodall, D. R., Zeros of chromatic polynomials, combinational surveys, (Cameron, P. J., Proc. 6th British Comb. Conf., (1977), Academic Press New York), 199-223 · Zbl 0357.05044
[14] Zykov, A. A., On some properties of linear complexes, Amer. Math. Soc. Transl., Math. Sb., 24, 66, 163-188, (1949)
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