OFFSET

4,1

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.

LINKS

Paolo Xausa, Table of n, a(n) for n = 4..10000

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.

Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.

N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report (2009).

FORMULA

a(n) = A351120(n-1) + 2

a(n) = ceiling((5 + sqrt(8*n - 7))/2) for n > 11.

EXAMPLE

The central vertex always requires two distinct colors. All vertices on the cycle require distinct pairs.

The colorings for small (broken) cycles 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

A350715[n_]:=If[n<12, {8, 8, 7, 7, 8, 7, 7, 8}[[n-3]], Ceiling[(5+Sqrt[8n-7])/2]]; Array[A350715, 100, 4] (* Paolo Xausa, Nov 30 2023 *)

CROSSREFS

KEYWORD

nonn

AUTHOR

Allan Bickle, Feb 02 2022

STATUS

approved