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 A381563 #4 Feb 28 2025 12:04:37 %S A381563 9,9,8,8,9,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,11,11,12, %T A381563 12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,14, %U A381563 14,14,14,15,15,15,15,15 %N A381563 2-tone chromatic number of a double wheel graph with n vertices. %C A381563 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 A381563 A double wheel has two vertices joined to a all vertices of a cycle. %H A381563 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 A381563 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 A381563 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. %F A381563 a(n) = A351120(n-2) + 3 = A350715(n-1) + 1. %F A381563 a(n) = ceiling((7 + sqrt(8*n - 15))/2) for n > 12. %e A381563 The central vertices share exactly one color. All vertices on the cycle require distinct pairs. %e A381563 The colorings for small (broken) cycles are shown below. %e A381563 -12-34-56- %e A381563 -12-34-15-36- %e A381563 -12-34-51-23-45- %e A381563 -12-34-15-32-14-35- %e A381563 -12-34-56-13-24-35-46- %e A381563 -12-34-15-23-14-25-13-45- %e A381563 -12-34-15-32-14-25-13-24-35- %Y A381563 Cf. A003057, A351120 (pair coloring). %Y A381563 Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar). %K A381563 nonn %O A381563 5,1 %A A381563 _Allan Bickle_, Feb 27 2025