

A309379


Number of unordered pairs of 4colorings of an nwheel 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
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OFFSET

3,1


COMMENTS

Please refer to the Wikipedia page on nwheel graphs linked here; an nwheel graph has n1 peripheral nodes and one central node, thus having n total nodes.


LINKS

Prateek Bhakta, Benjamin Brett Buckner, Lauren Farquhar, Vikram Kamat, Sara Krehbiel, Heather M. Russell, CutColorings in Coloring Graphs, Graphs and Combinatorics, (2019) 35(1), 239248.


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

Case n=4: The 4wheel 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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



