Let $n ≥ 3$. Define $G_n$ to be the graph where $V (G) = \{0, 1, . . . , n − 1\}$, and two vertices $a, b$ are adjacent if and only if $a ± 3 ≡ b (\mod n)$.
Graph Theory - Prove that Gn is connected if and only if $n$ is not a multiple of $3$.
0
$\begingroup$
group-theory
-
0just go from a = 1 by the law a->(a+3) mod n. – 2017-01-20
2 Answers
2
You can prove by induction that $a$ and $b$ are connected if and only if $a+3x\equiv b\pmod n$ for some integer $x$, that is, if and only if the congruence $$3x\equiv b-a\pmod n$$ has a solution. This congruence has a solution for all $a,b$ if and only if $\gcd(3,n)=1$, that is, $n$ is not a multiple of $3$.
0
HINT
Try to create an explicit cycle that walks through all vertices for $n$ not a multiple of $3$.
If the graph is not connected, then at least two vertices are not connected. Show you can't go from $1$ to $2$.
-
0Sorry, but I don't understand what you meant by "Show you can't go from 1 to 2." – 2017-01-20
-
0@Frank show there is no path from the vertex labeled $1$ to the vertex labeled $2$ – 2017-01-20