Question 82
Each vertex $v$ and each edge $e$ of a graph $G$ are assigned numbers $f(v) \in \{1, 2\}$ and $f(e) \in \{1, 2, 3\}$, respectively. Let $S(v)$ be the sum of numbers assigned to the edges incident to $v$ plus the number $f(v)$. We say that assignment $f$ is cool if $S(u) \neq S(v)$, for every pair of adjacent vertices in $G$. Prove that every graph has a cool assignment.