A351120 Pair chromatic number of a cycle graph with n vertices.

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

Offset: 3

Author

Allan Bickle, Feb 01 2022

Keywords

• nonn

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.

Examples

			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-
		

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.

Crossrefs

Cf. A003057, A350361, A350362.

Programs

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 *)

Formula

a(n) = ceiling((1 + sqrt(1 + 8*n))/2) for n > 10.