A350362 2-tone chromatic number of an n-cycle.

6, 6, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 3

Author

Allan Bickle, Dec 26 2021

Keywords

• nonn
• easy

Comments

The 2-tone 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 vertices at distance 2 have two common colors.

There is no 2-tone 5-coloring for cycles of length 3, 4, or 7 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-
Colorings for larger cycles can be spliced together from broken cycles of lengths 5, 6, and 8.
		

Links

• Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
• Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
• N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report, Group #2 Study Project, 2009.
• Index entries for eventually constant sequences
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A350361.

Programs

Mathematica

PadRight[{6,6,5,5,6},100,5] (* Paolo Xausa, Nov 30 2023 *)

Formula

a(n) = 5 for all n>7.

G.f.: x^3*(1 + x + x^4) + 5*x^3/(1 - x). - Stefano Spezia, Dec 27 2021