# Problem of the Month (December 2011)

Given a planar graph with no endpoints, what are the smallest non-negative labels the vertices can have so that all the inherited edge labels (which are the sum of the labels of the two vertices they connect) are different squares?

Here smallest means minimal total, and among those, minimal maximum vertex label.

Below are some graphs with 9 or fewer edges, and the smallest known vertex labelings. Can you improve any of these? Can you find any graphs I missed?

3 Edges

4 Edges

5 Edges
 (Andrew Bayly) (Andrew Bayly)

6 Edges
 (Jon Palin)

7 Edges

8 Edges
 (Jon Palin)

9 Edges
 (Jon Palin) (George Sicherman) (George Sicherman) (George Sicherman) (George Sicherman) (George Sicherman)

The smallest labels for cycles with n vertices are shown below:

Smallest Cycle Graphs with n Vertices
nlabelsauthor
30, 9, 16
40, 0, 9, 16
50, 1, 15, 21, 4Andrew Bayly
60, 0, 1, 15, 21, 4
70, 0, 1, 3, 22, 27, 9
80, 0, 1, 8, 8, 17, 32, 4
90, 0, 1, 8, 8, 17, 32, 32, 4George Sicherman
100, 0, 1, 8, 28, 21, 60, 61, 39, 25George Sicherman
110, 0, 1, 8, 8, 28, 21, 60, 61, 39, 25George Sicherman
120, 0, 1, 3, 6, 43, 38, 62, 59, 5, 11, 25George Sicherman
130, 0, 1, 3, 6, 10, 39, 61, 3, 22, 59, 85, 36George Sicherman
140, 0, 1, 3, 13, 51, 49, 32, 4, 5, 20, 101, 95, 49George Sicherman
150, 0, 1, 3, 6, 10, 26, 38, 11, 14, 67, 33, 88, 56, 169George Sicherman
160, 0, 1, 3, 6, 10, 111, 58, 6, 19, 17, 83, 113, 112, 32, 49Alex Rower
170, 0, 1, 3, 6, 10, 26, 55, 9, 16, 128, 128, 41, 8, 92, 104, 121Alex Rower
180, 0, 1, 3, 6, 10, 15, 34, 2, 62, 19, 81, 63, 162, 162, 34, 135, 121Alex Rower

The smallest labels for wheels with n vertices are shown below:

Smallest Wheel Graphs with n Vertices
ncenterothersauthor
423362, 359, 482Jon Palin
5194962, 62, 2, 482Jon Palin
62241712, 137, 32, 452, 2912Jon Palin
71443456, 25, 0, 256, 585, 640Jon Palin
8144880, 81, 0, 256, 585, 5040, 7956Jon Palin
9260701, 140, 29, 1340, 2141, 6140, 101, 524Jon Palin
10482962, 194, 2, 47, 1634, 1282, 9122, 3874, 887Jon Palin
11212364, 77, 44, 317, 2492, 1997, 1724, 877, 3884, 1157Jon Palin
128136, 188, 1928, 281, 248, 3473, 1288, 5768, 161, 568, 953George Sicherman

The smallest labels for n-dimensional cubes are shown below:

Smallest n-Dimensional Cube Graphs
nlabelsauthor
20, 0, 9, 16
30, 0, 9, 16, 153, 72, 49, 576George Sicherman

The smallest labels for complete graphs on n vertices are shown below:

Smallest Complete Graphs with n Vertices
nlabelsauthor
30, 9, 16
42, 359, 482, 3362Jon Palin
57442, 28658, 148583, 177458, 763442Jean-Louis Nicolas

In 2019, Alex Rower found these grid graphs, prism graphs, and complete bipartite graphs:

Smallest m×n Grid Graphs
m \ n2345
2 0 0
9 16
3 0  9
0 16
1 48
15 1 224
49 0 100
32  4  21
4 0  9
0 16
1 48
3 33
15     1 360
49     0 169
72     9   27
184 216   73
96   48   1 224
345   16   0 100
280     9   0 576
681 160 36 448
5 3     33
1     48
0     16
0       9
121 135
1   24 552
168   57 232
561   64 297
400     0 144
84   16 180
0 1296   385   291
0   225     99     70
121   504 1345 1155
75     25     24       1
825   264   760   840
0       0     1     24 1201
576     49 120     76   824
385     15 241   600 2425
1215 1921 288 1081     75
306   783     1 2400   409

Smallest Prism Graphs
nlabels
3   12   52   69
157 204 372
4     0   0   9   16
576 49 72 153
5     0     4   12   13   36
324 252 109 516 160
6   0     4   96   48   33   16
64 152 433 528 256 105

Smallest m×n Bipartite Graphs
m \ n234
2 0 9
0 16
3 1 49
0 15 120
0 273 768
16 256 1936
4 0 144
0 25 81 256
1 961 6241
0 483 1155 13923
?

Alex Rower also found cycle and grid graphs for triangular, pentagonal, hexagonal, and octagonal numbers.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/1/19.