This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350715 #19 Feb 28 2025 12:04:21
%S A350715 8,8,7,7,8,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,11,11,11,
%T A350715 11,11,11,11,11,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,
%U A350715 13,14,14,14,14,14,14,14,14,14,14,14,15,15,15,15,15
%N A350715 2-tone chromatic number of a wheel graph with n vertices.
%C A350715 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.
%H A350715 Paolo Xausa, <a href="/A350715/b350715.txt">Table of n, a(n) for n = 4..10000</a>
%H A350715 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 A350715 Allan Bickle, <a href="https://bica.the-ica.org/Volumes/103//Reprints/BICA2023-46-Reprint.pdf">2-Tone Coloring of Planar Graphs</a>, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
%H A350715 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 A350715 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> (2009).
%F A350715 a(n) = A351120(n-1) + 2
%F A350715 a(n) = ceiling((5 + sqrt(8*n - 7))/2) for n > 11.
%e A350715 The central vertex always requires two distinct colors. All vertices on the cycle require distinct pairs.
%e A350715 The colorings for small (broken) cycles are shown below.
%e A350715 -12-34-56-
%e A350715 -12-34-15-36-
%e A350715 -12-34-51-23-45-
%e A350715 -12-34-15-32-14-35-
%e A350715 -12-34-56-13-24-35-46-
%e A350715 -12-34-15-23-14-25-13-45-
%e A350715 -12-34-15-32-14-25-13-24-35-
%t A350715 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 *)
%Y A350715 Cf. A003057, A351120 (pair coloring).
%Y A350715 Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar), A381563 (double wheels).
%K A350715 nonn
%O A350715 4,1
%A A350715 _Allan Bickle_, Feb 02 2022