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

3,1

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.

Links

Table of n, a(n) for n=3..60.

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).

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

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-
Colorings for larger cycles can be spliced together from broken cycles of lengths 5, 6, and 8.

Mathematica

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

Crossrefs

Cf. A350361.

Sequence in context: A197478 A260713 A101801 * A351120 A199664 A019180

Adjacent sequences: A350359 A350360 A350361 * A350363 A350364 A350365

Keyword

nonn,easy

Author

Allan Bickle, Dec 26 2021

Status

approved