# Euler's formula for Planar Graph

I had a quite productive day yesterday when I flew from San Francisco to Tokyo, then travelled to Nagano because I read 50 pages of my Combinatorics book during my transit. I found one interesting theorem about planar graph so I’ll state here along with the proof: Euler’s formula: If $G=\left(V,E\right)$ is a connected planar graph with n (≥ 1) vertices, m edges, and r regions, then $n-m+r=2$.

## Proof by induction

### Basis

Consider the case $m=0$. In this case, n must be 1 since G is connected. With 1 vertex and no edges, there is exactly one outer region. Thus $n-m+r=1-0+1=2$ as desired.

### Inductive step

Assume that the statement is true for any connected planar graph with at most k edges. In other words, for any connected planar graph $G=\left(V,E\right)$ with n vertices $m\le k$, edges, and r regions, $n-m+r=2$. This is the inductive hypothesis.

Suppose $G=\left(V,E\right)$ is a connected planar graph with n vertices, $m=k+1$ vertices, and r regions. If G is a tree, then $m=n-1$ by the property of a tree. Notice that if G is a tree, then the only region is the outer region surround the tree so $r=1$. Thus $n-m+r=n-\left(n-1\right)+1=2$ as desired.

If G is not a tree, then G has a cycle. Call the cycle C. Let e be any edge in C, and consider a subgraph ${G}^{\mathrm{\prime }}=\left(V,E\setminus \left\{e\right\}\right)$ of G. ${G}^{\mathrm{\prime }}$ must be connected because removing an edge from a cycle breaks the cycle but does not break the connectivity of the graph. Furthermore, ${G}^{\mathrm{\prime }}$ has one fewer region than G because removing an edge merges exactly two regions into one region. Because a subgraph of a planar graph is also planar and $|E|=m=k+1$, ${G}^{\mathrm{\prime }}$ is a connected planar graph with $|E\setminus \left\{e\right\}|=k$ edges. Thus the inductive hypothesis applies to ${G}^{\mathrm{\prime }}$ and yields $n-k+\left(r-1\right)=2$. Therefore, $n-m+r=n-\left(k+1\right)+r=n-k+\left(r-1\right)=2$.

Thus in either case, $n-m+r=2$ as desired. Q.E.D.

## References

John M. Harris, J. L. H., & Mossinghoff, M. J. (2000). Combinatorics and Graph Theory. Springer.