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.
The square of a cycle is formed by adding edges between all vertices at distance 2 in the cycle.
LINKS
Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
Allan Bickle, 2-Tone Coloring of Chordal and Outerplanar Graphs, Australas. J. Combin. 87 1 (2023) 182-197.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
D. W. Cranston and H. LaFayette, The t-tone chromatic number of classes of sparse graphs, Australas. J. Combin. 86 (2023), 458-476.
N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report, Group #2 Study Project, 2009.
FORMULA
a(n) = 7 for all n>17.
EXAMPLE
The colorings for (broken) cycles with orders 7 through 13 are shown below.
-12-34-56-71-23-45-67-
-12-34-56-78-13-24-57-68-
-12-34-56-17-23-45-16-37-58-
-12-34-56-71-23-68-15-24-38-57-
-12-34-56-17-24-36-58-14-26-38-57-
-12-34-56-71-32-54-16-37-52-14-36-57-
-12-34-56-71-32-54-16-37-58-14-32-57-68-
CROSSREFS
KEYWORD
nonn
AUTHOR
Allan Bickle, Oct 17 2023
STATUS
approved