Aalok Sathe has authored 4 sequences.
A309379
Number of unordered pairs of 4-colorings of an n-wheel that differ in the coloring of exactly one vertex.
Original entry on oeis.org
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
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)
- 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
A309380
Number of unordered pairs of 5-colorings of an n-wheel that differ in the coloring of exactly one vertex.
Original entry on oeis.org
180, 240, 1380, 4200, 15420, 52080, 177780, 595320, 1978860, 6515520, 21298980, 69168840, 223369500, 717772560, 2296480980, 7319252760, 23247851340, 73615135200, 232462779780, 732245695080, 2301319648380, 7217727595440, 22594530691380, 70607719663800
Offset: 3
- Andrew Howroyd, Table of n, a(n) for n = 3..200
- 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
- Wikipedia, Wheel graph
- Index entries for linear recurrences with constant coefficients, signature (6,-6,-16,15,18).
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a(n) = {10*(2^(n-1) - 2*(-1)^n + (n-1)*(3^(n-2) - 3*(-1)^n))} \\ Andrew Howroyd, Sep 10 2019
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Vec(60*(3 - 14*x + 17*x^2 + 4*x^3 - 6*x^4)/((1 + x)^2*(1 - 2*x)*(1 - 3*x)^2) + O(x^30)) \\ Andrew Howroyd, Sep 10 2019
A307334
Number of 3-colorings of an n-dimensional hypercube.
Original entry on oeis.org
3, 6, 18, 114, 2970, 1185282, 100301050602
Offset: 0
- 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.
- Eric Weisstein's World of Mathematics, Hypercube Graph
- Eric Weisstein's World of Mathematics, Vertex Coloring
- Wikipedia, Chromatic polynomial
- Wikipedia, Hypercube
A309315
Number of 5-colorings of an n-wheel graph.
Original entry on oeis.org
60, 120, 420, 1200, 3660, 10920, 32820, 98400, 295260, 885720, 2657220, 7971600, 23914860, 71744520, 215233620, 645700800, 1937102460, 5811307320, 17433922020, 52301766000, 156905298060, 470715894120, 1412147682420, 4236443047200, 12709329141660
Offset: 3
- Colin Barker, Table of n, a(n) for n = 3..1000
- 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.
- Eric Weisstein's World of Mathematics, Wheel Graph
- Wikipedia, Chromatic polynomial
- Wikipedia, Wheel graph
- Index entries for linear recurrences with constant coefficients, signature (2,3).
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