I have a question regarding a graph theory problem, and I'm seeking assistance in understanding the provided solution. I comprehend that the solution needs to establish reflexivity, antisymmetry, and transitivity, but I'm struggling to interpret the proof steps.
The problem: Let $G=(V, E)$ be a connected, loop-free graph. Furthermore, assume that $G$ has 53 faces. For any planar representation of $G$, each subregion is bounded by at least five edges. I need to prove that $|V| \geq 82$.
Solution: The solution starts by introducing $G=(E, V)$ and defining the number of edges as $e=|E|$ and the number of vertices as $v=|V|$. It also mentions the number of faces as $r$. Then, it employs the concept that edges contribute to two faces to derive the inequality $2e \geq 5 \times r \Leftrightarrow e \geq 5 \frac{r}{2}$.
Using Euler's polyhedral formula $v-e+r=2,$ the solution continues to show that $v \geq 81.5$. It concludes that the number of vertices is 82 or more.
However, I'm having trouble following the logical steps in this proof. I need help breaking down the logic behind this proof and understanding how it establishes that $|V| \geq 82$.