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 A309379 Number of unordered pairs of 4-colorings of an n-wheel that differ in the coloring of exactly one vertex. 2
 36, 0, 108, 120, 444, 840, 2124, 4536, 10332, 22440, 49260, 106392, 229500, 491400, 1048716, 2228088, 4718748, 9961320, 20971692, 44040024, 92274876, 192937800, 402653388, 838860600, 1744830684, 3623878440, 7516193004, 15569256216, 32212254972, 66571992840 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS Please refer to the Wikipedia page on n-wheel graphs linked here; an n-wheel graph has n-1 peripheral nodes and one central node, thus having n total nodes. LINKS Table of n, a(n) for n=3..32. Prateek Bhakta, Benjamin Brett Buckner, Lauren Farquhar, Vikram Kamat, Sara Krehbiel, Heather M. Russell, Cut-Colorings in Coloring Graphs, Graphs and Combinatorics, (2019) 35(1), 239-248. Luis Cereceda, Janvan den Heuvel, Matthew Johnson, Connectedness of the graph of vertex-colourings, Discrete Mathematics, (2008) 308(5-6), 913-919. Aalok Sathe, Coloring Graphs Library Aalok Sathe, iPython notebook to generate terms of this sequence Wikipedia, Wheel graph FORMULA G.f.: 12*x^3*(3 - 9*x + 6*x^2 + 4*x^3 - 2*x^4)/((1 - x)*(1 + x)^2*(1 - 2*x)^2). - Andrew Howroyd, Aug 27 2019 EXAMPLE From Andrew Howroyd, Aug 27 2019: (Start) Case n=4: The 4-wheel graph is isomorphic to the complete graph on 4 vertices. Each vertex must be colored differently and it is not possible to change the color of just one vertex and still leave a valid coloring, so a(4) = 0. Case n=5: The peripheral nodes can colored using one of the patterns 1212, 1213 or 1232. In the case of 1212, colors can be selected in 24 ways and any vertex including the center vertex can be flipped to the unused color giving 24*5 = 120. In the case of 1213 or 1232, colors can be selected in 24 ways and two vertices can have a color change giving 24*2*2 = 96. Since we are counting unordered pairs, a(5) = (120 + 96)/2 = 108. (End) PROG (PARI) Vec(12*(3 - 9*x + 6*x^2 + 4*x^3 - 2*x^4)/((1 - x)*(1 + x)^2*(1 - 2*x)^2) + O(x^30)) \\ Andrew Howroyd, Aug 27 2019 CROSSREFS Cf. A090860 (number of 4-colorings), A309315 (number of 5-colorings), A309380. Sequence in context: A292156 A265489 A262795 * A022071 A181635 A174673 Adjacent sequences: A309376 A309377 A309378 * A309380 A309381 A309382 KEYWORD nonn AUTHOR Aalok Sathe, Jul 26 2019 EXTENSIONS Terms a(14) and beyond from Andrew Howroyd, Aug 27 2019 STATUS approved

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