The Degree Diameter Problem for Planar Graphs

From Combinatorics Wiki

Jump to: navigation, search

Contents

Table of the orders of the largest known regular planar graphs for the undirected degree diameter problem

d\k 2 3 4 5 6 7 8
3 6 12 18 28 36 52 76
4 9 16 27 44 81 134 243
5 NA 16 28 62 124 254 500

The following table is the key to the colors in the table presented above:

Color Details
* Graphs found by Preen.
* Graphs found by Pratt and Friedman.

Table of the lowest upper bounds known at present, and the percentage of the order of the largest known regular planar graphs

d\k 2 3 4 5 6 7 8
3
6
100%
12
100%
30
60%
62
45.16%
122
29.5%
244
21.31%
488
15.57%
4
9
100%
33
48.48%
96
28.13%
291
15.12%
867
9.34%
2595
5.16%
7779
3.12%
5 NA
16
100%
248
11.29%
984
6.3%
3928
3.16%
11294
2.25%
62808
0.79%


The following table is the key to the colors in the table presented above:

Color Details
* Bound by Preen.

Table of the orders of the largest known planar graphs for the undirected degree diameter problem

d\k 2 3 4 5 6 7 8 9 10
3 7 12 18 28 38 53 77109157
4 9 16 27 44 81134 243404728
5 10 19 39 73 15828963811532558
6 11 24 55 114 280564140528147030
7 12 28 74 161 452 9592720574716328
8 13 33 97 225 685 156949011097733613
9 14 37 122 289 986 230578981843363194
10 16 42 151 372 1366 33421230130072110716

The following table is the key to the colors in the table presented above:

Color Details
* Graphs found by Fellows, Hell, and Seyffarth.
* Graphs found by Yang, Lin, and Dai.
* Graphs found by Geoffrey Exoo.
* Graphs found by S. A. Tishchenko.

Table of the lowest upper bounds known at present, and the percentage of the order of the largest known graphs

d\k 2 3 4 5 6
3
7
100%
15
80%
31
58.06%
63
44.44%
127
29.92%
4
9
100%
35
45.71%
104
25.96%
313
14.06%
934
8.67%
5
10
100%
52
36.54%
248
15.73%
984
7.42%
3928
3.82%
6
11
100%
60
40%
521
9.98%
2409
4.73%
12971
2.02%
7
12
100%
68
41.18%
938
7.57%
3267
4.93%
26793
1.59%
8
13
100%
76
43.42%
1529
6.08%
4257
5.28%
39975
1.63%
9
14
100%
84
44.05%
2324
5.08%
5379
5.37%
56901
1.66%
10
16
100%
92
45.65%
3353
4.35%
6633
5.61%
78039
1.68%


The following table is the key to the colors in the table presented above:

Color Details
* Bound by Fellows, Hell, and Seyffarth.
* Bound by Yang, Yuansheng, Lin, Jianhua, Dai, Yongjie.

External links