OFFSET
1,1
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.
a(n) is also the 2-tone chromatic number of a star with n leaves.
LINKS
A. 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, (2009).
FORMULA
a(n) = A003057(n-1) + 2.
a(n) = ceiling((5 + sqrt(1 + 8*n))/2).
EXAMPLE
For a star with three leaves, label the leaves 12, 13, and 23. Label the other vertex 45. A total of 5 colors are used, so a(3)=5.
MATHEMATICA
Table[Ceiling[(5 + Sqrt[1 + 8*n])/2], {n, 71}] (* Stefano Spezia, Dec 27 2021 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Dec 26 2021
STATUS
approved