cp's OEIS Frontend

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.

Showing 1-7 of 7 results.

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

Original entry on oeis.org

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

Views

Author

Allan Bickle, Feb 02 2022

Keywords

Comments

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.

Examples

			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-

Links

Crossrefs

Cf. A003057, A351120 (pair coloring).
Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar), A381563 (double wheels).

Programs

  • Mathematica
    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 *)

Formula

a(n) = A351120(n-1) + 2
a(n) = ceiling((5 + sqrt(8*n - 7))/2) for n > 11.

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

Original entry on oeis.org

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, 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, 15, 15
Offset: 1

Views

Author

Allan Bickle, Oct 17 2023

Keywords

Comments

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.
a(n) is also the 2-tone chromatic number of a fan with n+1 vertices.

Examples

			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.

Links

Crossrefs

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366728 (cycle squared).
Cf. A003057, A351120 (pair coloring).

Formula

a(n) = ceiling(sqrt(2*n + 1/4) + 5/2) for n > 6.

A366728 2-tone chromatic number of the square of a cycle with n vertices.

Original entry on oeis.org

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, 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, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
Offset: 3

Views

Author

Allan Bickle, Oct 17 2023

Keywords

Comments

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

Examples

			The colorings for (broken) cycles with orders 7 through 13 are shown below.
  -12-34-56-71-23-45-67-
  -12-34-56-78-13-24-57-68-
  -12-34-56-17-23-45-16-37-58-
  -12-34-56-71-23-68-15-24-38-57-
  -12-34-56-17-24-36-58-14-26-38-57-
  -12-34-56-71-32-54-16-37-52-14-36-57-
  -12-34-56-71-32-54-16-37-58-14-32-57-68-

Links

Crossrefs

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (MOPs).
Cf. A003057, A351120 (pair coloring).

Formula

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

A381562 Minimum 2-tone chromatic number of maximal planar graphs with n vertices.

Original entry on oeis.org

6, 8, 9, 9, 8, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 7
Offset: 3

Views

Author

Allan Bickle, Feb 27 2025

Keywords

Comments

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.
For n in {19,22,23,27}, a(n) is either 7 or 8. All larger values are 7.

Examples

			For n=3, all 3 vertices get two distinct colors, so a(3) = 6.
For n=4, all 4 vertices get two distinct colors, so a(3) = 8.
For n=5 or 6, the extremal graph is a double wheel.

Links

Crossrefs

Cf. A003057, A351120 (pair coloring).
Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles).

Formula

a(n) = 7 for n > 27.

A381563 2-tone chromatic number of a double wheel graph with n vertices.

Original entry on oeis.org

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, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15
Offset: 5

Views

Author

Allan Bickle, Feb 27 2025

Keywords

Comments

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.
A double wheel has two vertices joined to a all vertices of a cycle.

Examples

			The central vertices share exactly one color.  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-

Links

Crossrefs

Cf. A003057, A351120 (pair coloring).
Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar).

Formula

a(n) = A351120(n-2) + 3 = A350715(n-1) + 1.
a(n) = ceiling((7 + sqrt(8*n - 15))/2) for n > 12.

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

Views

Author

Allan Bickle, Feb 27 2025

Keywords

Comments

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 graphs are maximal planar.

Examples

			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

Links

Crossrefs

Cf. A003057, A351120 (pair coloring).
Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar), A381563 (double wheels).

Formula

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

A381565 2-tone chromatic number of a particular class of planar graphs with 3n+3 vertices.

Original entry on oeis.org

5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21
Offset: 1

Views

Author

Allan Bickle, Feb 27 2025

Keywords

Comments

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 graphs are formed by replacing each edge of K_3 by n disjoint paths with length 2, resulting in 3n+3 vertices. These graphs have large 2-tone chromatic number relative to their maximum degree of 2t.

Examples

			For n=1, the graph is a 6-cycle, which has a 2-tone 5-coloring -12-34-15-32-14-35-.  Thus a(1) = 5.

Links

Crossrefs

Cf. A003057, A351120 (pair coloring).
Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar).

Formula

a(n) = ceiling(1.5 + sqrt(6*n + 6.25)) for n < 18.
a(n) = ceiling(0.5 + sqrt(6*n + 24.25)) for n > 6.
Showing 1-7 of 7 results.

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