OFFSET
3,1
COMMENTS
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.
LINKS
Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer., Vol. 217 (2013), pp. 171-190.
Allan Bickle and Ben Phillips, t-Tone Colorings of Graphs, Utilitas Math, Vol. 106 (2018), pp. 85-102.
FORMULA
a(n) = ceiling((1 + sqrt(1 + 8*n))/2) for n > 10.
EXAMPLE
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-
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Allan Bickle, Feb 01 2022
STATUS
approved