A381563 -id:A381563 - cp's OEIS frontend

A350715 2-tone chromatic number of a wheel graph with n vertices.

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, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15

Offset: 4

Allan Bickle, Feb 02 2022

nonn

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 central vertex always requires two distinct colors.  All vertices on the cycle require distinct pairs.
The colorings for small (broken) cycles are shown below.
  -12-34-56-
  -12-34-15-36-
  -12-34-51-23-45-
  -12-34-15-32-14-35-
  -12-34-56-13-24-35-46-
  -12-34-15-23-14-25-13-45-
  -12-34-15-32-14-25-13-24-35-

Paolo Xausa, Table of n, a(n) for n = 4..10000
Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
Allan Bickle, 2-Tone Coloring of Planar Graphs, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report (2009).

Cf. A003057, A351120 (pair coloring).

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar), A381563 (double wheels).

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 *)

a(n) = A351120(n-1) + 2

a(n) = ceiling((5 + sqrt(8*n - 7))/2) for n > 11.

A381564 2-tone chromatic number of a path with n-2 vertices joined to two adjacent vertices.

Original entry on oeis.org

8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 4

Allan Bickle, Feb 27 2025

nonn

The graphs are maximal planar.

			The central vertices each have two disjoint labels.  All vertices on the path require distinct pairs.
The colorings for small paths are shown below.
  12-34
  12-34-15
  12-34-15-23
  12-34-15-23-14
  12-34-15-23-14-25
  12-34-15-23-14-25-13
  12-34-15-23-14-25-13-24
  12-34-15-23-14-25-13-24-35

Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013) 171-190.
Allan Bickle, 2-Tone Coloring of Planar Graphs, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.

Cf. A003057, A351120 (pair coloring).

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar), A381563 (double wheels).

a(n) = ceiling((9 + sqrt(8*n - 15))/2) for n > 8.

cp's OEIS Frontend

A350715 2-tone chromatic number of a wheel graph with n vertices.

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