MathDB

In the figures, the vertices are marked with a circle. The segments that join vertices are called paths. Non-negative integers are distributed to the vertices and, to the paths, the differences between the numbers at their ends. https://cdn.artofproblemsolving.com/attachments/d/6/e6fce93719a5b35dbf34d58652b01a8631de57.gif
We will say that a distribution of numbers is graceful if all the numbers from

1

n

appear in the paths, where

n

is the number of paths.
The following is an example of graceful distribution: https://cdn.artofproblemsolving.com/attachments/1/1/a8c2b4fde673ca902b655804c4f5321f9666e9.gif Give -if possible- a graceful distribution for the following figures. If you can't do it, show why.

Problem Statement