# Large Regular Planar Graphs

# With Small Diameter

#### A n-regular graph is a graph in which every vertex has degree n . The diameter of a graph is the length of the longest distance in the graph. What is the largest number of vertices a planar n-regular graph with diameter d can have? Here are the best known bounds for small n and d . Click on the links for pictures.

Largest planar n-regular graph with diameter d

n \ d |
1 |
2 |
3 |
4 |
5 |
6 |

1 |
2 |
none |
none |
none |
none |
none |

2 |
3 |
5 |
7 |
9 |
11 |
13 |

3 |
4 |
6 |
12 |
18, 30 |
28, 62 |
36, 122 |

4 |
none |
9 |
16, 33 |
27, 96 |
44, 291 |
81, 867 |

5 |
none |
none |
16, 52 |
28, 248 |
62, 984 |
124, 3928 |