cp's OEIS Frontend

A309379 Number of unordered pairs of 4-colorings of an n-wheel that differ in the coloring of exactly one vertex.

%I A309379 #26 Aug 28 2019 12:11:20
%S A309379 36,0,108,120,444,840,2124,4536,10332,22440,49260,106392,229500,
%T A309379 491400,1048716,2228088,4718748,9961320,20971692,44040024,92274876,
%U A309379 192937800,402653388,838860600,1744830684,3623878440,7516193004,15569256216,32212254972,66571992840
%N A309379 Number of unordered pairs of 4-colorings of an n-wheel that differ in the coloring of exactly one vertex.
%C A309379 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.
%H A309379 Prateek Bhakta, Benjamin Brett Buckner, Lauren Farquhar, Vikram Kamat, Sara Krehbiel, Heather M. Russell, <a href="https://doi.org/10.1007/s00373-018-1985-6">Cut-Colorings in Coloring Graphs</a>, Graphs and Combinatorics, (2019) 35(1), 239-248.
%H A309379 Luis Cereceda, Janvan den Heuvel, Matthew Johnson, <a href="https://doi.org/10.1016/j.disc.2007.07.028">Connectedness of the graph of vertex-colourings</a>, Discrete Mathematics, (2008) 308(5-6), 913-919.
%H A309379 Aalok Sathe, <a href="https://github.com/aalok-sathe/coloring-graphs">Coloring Graphs Library</a>
%H A309379 Aalok Sathe, <a href="/A309379/a309379.ipynb.txt">iPython notebook to generate terms of this sequence</a>
%H A309379 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wheel_graph">Wheel graph</a>
%F A309379 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
%e A309379 From _Andrew Howroyd_, Aug 27 2019: (Start)
%e A309379 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.
%e A309379 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.
%e A309379 (End)
%o A309379 (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
%Y A309379 Cf. A090860 (number of 4-colorings), A309315 (number of 5-colorings), A309380.
%K A309379 nonn
%O A309379 3,1
%A A309379 _Aalok Sathe_, Jul 26 2019
%E A309379 Terms a(14) and beyond from _Andrew Howroyd_, Aug 27 2019