%I #15 Nov 30 2023 08:46:10

%S 6,8,10,9,7,8,8,8,8,7,8,7,7,7,8,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,

%T 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,

%U 7,7,7,7,7,7,7,7,7,7,7,7,7,7,7

%N 2-tone chromatic number of the square of a cycle with n vertices.

%C 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.

%C The square of a cycle is formed by adding edges between all vertices at distance 2 in the cycle.

%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/2tonejcpaper.pdf">2-Tone coloring of joins and products of graphs</a>, Congr. Numer. 217 (2013), 171-190.

%H Allan Bickle, <a href="https://ajc.maths.uq.edu.au/pdf/87/ajc_v87_p182.pdf">2-Tone Coloring of Chordal and Outerplanar Graphs</a>, Australas. J. Combin. 87 1 (2023) 182-197.

%H Allan Bickle and B. Phillips, <a href="https://allanbickle.files.wordpress.com/2016/05/ttonepaperb.pdf">t-Tone Colorings of Graphs</a>, Utilitas Math, 106 (2018) 85-102.

%H D. W. Cranston and H. LaFayette, <a href="https://ajc.maths.uq.edu.au/pdf/86/ajc_v86_p458.pdf">The t-tone chromatic number of classes of sparse graphs</a>, Australas. J. Combin. 86 (2023), 458-476.

%H N. Fonger, J. Goss, B. Phillips, and C. Segroves, <a href="https://web.archive.org/web/20220121030248/https://homepages.wmich.edu/~zhang/finalReport2.pdf">Math 6450: Final Report</a>, Group #2 Study Project, 2009.

%F a(n) = 7 for all n>17.

%e The colorings for (broken) cycles with orders 7 through 13 are shown below.

%e -12-34-56-71-23-45-67-

%e -12-34-56-78-13-24-57-68-

%e -12-34-56-17-23-45-16-37-58-

%e -12-34-56-71-23-68-15-24-38-57-

%e -12-34-56-17-24-36-58-14-26-38-57-

%e -12-34-56-71-32-54-16-37-52-14-36-57-

%e -12-34-56-71-32-54-16-37-58-14-32-57-68-

%Y Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (MOPs).

%Y Cf. A003057, A351120 (pair coloring).

%K nonn

%O 3,1

%A _Allan Bickle_, Oct 17 2023