I know with the handshake Lemma that the sum of all degrees of the 3 regular graph is equal to 2 x amount of edges.
so: $\sum_{v \in V}: deg(v) = 2 \cdot |E|$
for every even number $n\geq 4$, is $n = 2p, p\geq 2$,
$$\sum_{v \in V} deg(v) = 2 \cdot |E| \iff 3n = 2* |E|\\ \iff 3*2p = 2*|E| \iff 3*2p = 2*3p$$
If we set p = 2,3,4.. we see clearly that the equation 3*2p = 2*3p is true for every p. Is it already proven or do I have to prove it by using induction on n with 3*2p = 2*3p ? If not, how can I prove it? Or do I just have to draw a graph like here: Prove that for every even $n\ge 4 $, there exists a planar, connected and 3-regular graph with $n$ vertices ?
Best Answer
Construct the cycle $C_{2n}=(1,2,...,2n,1)$ on vertices $\{1,2,...,2n\}$ then connect vertex $i$ to $i+n$ for $i \in \{1,2,...,n\}$.