Abstract
With an arbitrary graph G having n vertices and m edges, and with an arbitrary natural number p, we associate in a natural way a polynomial R(x 1,...,x n) with integer coefficients such that the number of colorings of the vertices of the graph G in p colors is equal to p m-n R(0,...,0). Also with an arbitrary maximal planar graph G, we associate several polynomials with integer coefficients such that the number of colorings of the edges of the graph G in 3 colors can be calculated in several ways via the coefficients of each of these polynomials. Bibliography: 2 titles.
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REFERENCES
Yu. Matiyasevich, “Problem 21,” Combinatorial Asymptotical Analisys, Vol. 2, Krasnoyarskii State University, Krasnoyarsk (1977), pp. 178–179.
Yu. Matiyasevich, “A polynomial related to colourings of triangulation of sphere,” URL: http://www. informatik. uni-stuttgart/de/ifi/ti/personen/ Matiyasevich/Journal/Triangular/triang.htm.
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Matiyasevich, Y.V. Some Algebraic Methods for Calculating the Number of Colorings of a Graph. Journal of Mathematical Sciences 121, 2401–2408 (2004). https://doi.org/10.1023/B:JOTH.0000024621.54839.40
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DOI: https://doi.org/10.1023/B:JOTH.0000024621.54839.40