Let $C$ be a Hamiltonian cycle on a graph with vertices labeled {$1,...,9$}. Prove that there are $3$ vertices adjacent in $C$ whose labels sum to at least $12$.
I understand why this fact is true by drawing a graph and different and calculating the adjacent vertices sum to be at least $12$.
How would I go about proving it? Would I use the definition of a Hamiltonian cycle and how would I show that the vertices have a minimum value of $12$?
Consider the $9$ vertex. The minimum sum it can be a part of is $1 + 2 + 9 = 12$. Hence the sum of the vertices around $9$ (and including it) is always at least $12$.