Eulers Identity: n-m+r=2
a)If G is a planar graph which contains no cycles of length less than g, then give an improved version of Eulers identity by relating the number of regions to the number of edges.
b)use (a) to show Petersen graph is not planar
I'm thinking i need to relate n with g such that a new euler identity can be written with an inequality instead...not sure how they relate though.
Hint 1: Assume by contradiction it is planar.
Since you know $n,m$ by Euler you get $r$.
Hint 2 In the Petersen graph, If you count the edges by faces, each of the $r$ faces has at least $5$ edges. So your count is at least $5r$.
In this count each edge was counted exactly twice. So your count is exactly $2m$.
This shows $$2m \geq 5r$$
See what you get.