Problem of the Month (September 2019)

Given a positive integer N, consider connected planar graphs whose vertices can be labeled UNIQUELY with positive integers so that the labels of the neighbors of every vertex sum to N. Among those graphs, what is the smallest graph with labels S, some subset of positive integers?

ANSWERS

N=1
{1}

N=2
{1}
{2}
{1,2}

N=3
{1}
{1,2}
{3}
{1,3}

N=4
{1}

(George Sicherman) 		{2}
{1,2}
{1,3}
{1,2,3}
{4}
{1,4}

N=5
{1}

(George Sicherman) 		{1,2}
{1,3}
{1,2,3}
{1,4}

(George Sicherman)
{1,2,4}

(George Sicherman) 		{1,3,4}
{1,2,3,4}
{5}
{1,5}

N=6
{2}
{1,2}
{3}
{1,3}
{1,2,3}
{1,4}
{2,4}
{1,2,4}
{1,3,4}
{2,3,4}

(Joe DeVincentis) 		{1,2,3,4}
{1,5}
{1,2,5}

(George Sicherman) 		{1,3,5}
{1,2,3,5}
{1,4,5}
{1,2,4,5}
{1,3,4,5}
{1,2,3,4,5}
{6}
{1,6}

N=7
{1,2}

(George Sicherman) 		{1,3}
{1,2,3}
{1,4}

(Torsten Ueckerdt) 		{1,2,4}

(George Sicherman) 		{1,3,4}
{1,2,3,4}
{1,5}
{1,2,5}
{1,3,5}

(Berend van der Zwaag) 		{1,2,3,5}
{1,4,5}

(Berend van der Zwaag) 		{1,2,4,5}
{1,3,4,5}

(George Sicherman)
{1,2,3,4,5}
{1,6}

(George Sicherman) 		{1,2,6}

(Berend van der Zwaag) 		{1,3,6}
{1,2,3,6}

(George Sicherman) 		{1,4,6}

(Joe DeVincentis) 		{1,2,4,6}

(George Sicherman)
{1,3,4,6}
{1,2,3,4,6}
{1,5,6}
{1,2,5,6}
{1,3,5,6}

(George Sicherman) 		{1,2,3,5,6}
{1,4,5,6}

(George Sicherman) 		{1,2,4,5,6}
{1,3,4,5,6}

(George Sicherman) 		{1,2,3,4,5,6}
{7}
{1,7}

N=8
{2}

(George Sicherman) 		{1,2}

(George Sicherman) 		{1,3}

(Joe DeVincentis) 		{2,3}
{1,2,3}
{4}
{1,4}
{2,4}

(George Sicherman)
{1,2,4}
{1,3,4}
{2,3,4}
{1,2,3,4}
{1,5}

(Joe DeVincentis) 		{1,2,5}

(George Sicherman) 		{1,3,5}

(George Sicherman)
{2,3,5}
{1,2,3,5}
{1,4,5}

(George Sicherman) 		{1,2,4,5}
{1,3,4,5}

(George Sicherman) 		{2,3,4,5}
{1,2,3,4,5}
{1,6}
{2,6}
{1,2,6}
{1,3,6}

(George Sicherman) 		{2,3,6}

(Joe DeVincentis) 		{1,2,3,6}

(George Sicherman) 		{1,4,6}

(George Sicherman) 		{2,4,6}
{1,2,4,6}

(George Sicherman) 		{1,3,4,6}

(George Sicherman)
{2,3,4,6}

(George Sicherman) 		{1,2,3,4,6}
{1,5,6}

(George Sicherman) 		{1,2,5,6}

(George Sicherman) 		{1,3,5,6}

(George Sicherman) 		{2,3,5,6}

(George Sicherman) 		{1,2,3,5,6}
{1,4,5,6}

(George Sicherman)
{1,2,4,5,6}

(George Sicherman) 		{1,3,4,5,6}

(George Sicherman) 		{2,3,4,5,6}

(George Sicherman) 		{1,2,3,4,5,6}

(George Sicherman) 		{1,7}
{1,2,7}

(George Sicherman) 		{1,3,7}

(George Sicherman) 		{1,2,3,7}

(George Sicherman)
{1,4,7}
{1,2,4,7}
{1,3,4,7}

(George Sicherman) 		{1,2,3,4,7}

(George Sicherman) 		{1,5,7}

(George Sicherman) 		{1,2,5,7}

(George Sicherman) 		{1,3,5,7}

(George Sicherman)
{1,2,3,5,7}
{1,4,5,7}

(George Sicherman) 		{1,2,4,5,7}

(George Sicherman) 		{1,3,4,5,7}

(George Sicherman) 		{1,2,3,4,5,7}

(George Sicherman) 		{1,6,7}

(George Sicherman) 		{1,2,6,7}
{1,3,6,7}

(George Sicherman)
{1,2,3,6,7}

(George Sicherman) 		{1,4,6,7}

(George Sicherman) 		{1,2,4,6,7}
{1,3,4,6,7}

(George Sicherman) 		{1,2,3,4,6,7}

(George Sicherman) 		{1,5,6,7}

(George Sicherman)
{1,2,5,6,7}

(George Sicherman) 		{1,3,5,6,7}

(George Sicherman) 		{1,2,3,5,6,7}
{1,4,5,6,7}

(George Sicherman) 		{1,2,4,5,6,7}

(George Sicherman) 		{1,3,4,5,6,7}

(George Sicherman) 		{1,2,3,4,5,6,7}

(George Sicherman) 		{8}
{1,8}

Torsten Ueckerdt proved that N=6 {2,3}, N=7 {2,3}, and N=7 {2,3,4} have no solutions. Joe DeVincentis proved that N=7 {2,3,4,5} has no solution. Bryce Herdt proved that N=8 {2,5,6} has no solution.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 9/1/19.