cp's OEIS Frontend

A366727 2-tone chromatic number of a maximal outerplanar graph with maximum degree n.

%I A366727 #16 Nov 30 2023 07:31:03
%S A366727 4,6,7,7,7,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,11,11,
%T A366727 11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,
%U A366727 13,13,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15
%N A366727 2-tone chromatic number of a maximal outerplanar graph with maximum degree n.
%C A366727 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 A366727 a(n) is also the 2-tone chromatic number of a fan with n+1 vertices.
%H A366727 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 A366727 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 A366727 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 A366727 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 A366727 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 A366727 a(n) = ceiling(sqrt(2*n + 1/4) + 5/2) for n > 6.
%e A366727 The fan with 11 vertices has a path colored 12-34-15-23-45-13-24-35-14-25 joined to a vertex colored 67, so a(10) = 7.
%Y A366727 Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366728 (cycle squared).
%Y A366727 Cf. A003057, A351120 (pair coloring).
%K A366727 nonn
%O A366727 1,1
%A A366727 _Allan Bickle_, Oct 17 2023