I will solve a graph theory problem:

Let G be a connected 3-regular plane graph in which every vertex lies on one face of length 4, one face of length 6, and one face of length 8.

(a) In terms of n(G), determine the number of faces of each length.

(b) Use Euler’s Formula and part(a) to determine the number of faces of G.

## Graph Theory Problem Solving:

Let $e=e(G)$, $n=n(G)$ and $f$ be the number of faces from the graph $G$. Since $G$ is a 3-regular graph, $2e=3n$. For each $i \in \{4,6,8\}$, let $f_i$ be the number of faces whose length is $i$. Each vertex lies on one face of length $i$, for all $i\in \{4,6,8\}$. Therefore $n=if_i$ for all $i \in \{ 4,6,8\}$. Therefore, $f_i = \frac{n}{i}$ for all $i \in \{4,6,8\}$. By Euler’s formula $n-e+f=2$. Therefore, $n- \frac{3}{2} n + \sum_{i \in \{4,6,8\}} n/ i =2$. Therefore, $n=48$. From the relation $n=if_i$, $f_4=12$, $f_6=8$ and $f_8=6$.

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Source: Homework problem from Professor Choi‘s Graph Theory class, Gwangju Institute of Science and Technology, Fall Semester 2023