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A351120
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Pair chromatic number of a cycle graph with n vertices.
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4
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6, 6, 5, 5, 6, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14
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OFFSET
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3,1
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COMMENTS
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The pair chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of colors is repeated.
There is no pair 5-coloring for cycles of length 3, 4, 7, or 10 since the Petersen graph does not contain cycles of these lengths.
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LINKS
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FORMULA
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a(n) = ceiling((1 + sqrt(1 + 8*n))/2) for n > 10.
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EXAMPLE
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The colorings for (broken) cycles with orders 3 through 9 are shown below.
-12-34-56-
-12-34-15-36-
-12-34-51-23-45-
-12-34-15-32-14-35-
-12-34-56-13-24-35-46-
-12-34-15-23-14-25-13-45-
-12-34-15-32-14-25-13-24-35-
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MATHEMATICA
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A351120[n_]:=If[n<11, {6, 6, 5, 5, 6, 5, 5, 6}[[n-2]], Ceiling[(1+Sqrt[1+8n])/2]]; Array[A351120, 100, 3] (* Paolo Xausa, Nov 30 2023 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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