MathDB

Let

G

be a graph whose vertices are the integers. Assume that any two integers are connected by a finite path in

G

. For two integers

x

and

y

, we denote by

d(x, y)

the length of the shortest path from

x

y

, where the length of a path is the number of edges in it. Assume that

d(x, y) \mid x-y

for all integers

x, y

and define

S(G)=\{d(x, y) | x, y \in \mathbb{Z}\}

. Find all possible sets

S(G)

9