A309379 Number of unordered pairs of 4-colorings of an n-wheel that differ in the coloring of exactly one vertex.
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
Offset: 3
Keywords
Examples
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)
Links
- 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
Programs
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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
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
Extensions
Terms a(14) and beyond from Andrew Howroyd, Aug 27 2019
Comments