Is Euler's formula in planar graphs proved not using induction? If we have G as a connected planar graph or a multi graph with |V|=v and |E|=e and r as the number of regions in the planar graph G.Then according to Euler's formula we have v-e+r=2.It has been proved by using induction method.
Is there  another way to prove the Euler's formula? 
 A: Yes. You can use algebraic topology to compute the Euler characteristic of the CW-complex represented by the graph (which you embed in the sphere by "adding a point at infinity"). Since the resulting CW-complex is topologically a sphere, it must have the same euler characteristic (vertices - edges + faces) as the sphere, i.e., 2. 
I know that's not a very satisfactory answer, since you probably don't know many of those words, but that is a valid (and interesting!) alternative approach. 
A: Yes, you can embedd the planar graph into a sphere (WLOG, into a sphere without the south-pole) and then the graph determines a cell-complex structure on the sphere. That is, vertices are vertices of the graph, $1$-cells are edges, and $2$-cells are maps that map the 2-balls to the "regions" bounded by their boundary circles. The unbounded region will be mapped to that component on the sphere that contains the "infinity" (south pole, for example). It is a common fact in algebraic topology that the Euler number of the 2-sphere is $2$ and can be computed this way (Nr of vertices minus Nr of edges...) for any triangulation resp. for any cell structure. 
This is not as abstract as it may look like. If you can draw a graph onto a torus, than you (usually) have that the number of vertices minus the number of edges plus the number of regions (in the torus) is zero (which is the Euler characteristic of the torus). Here is an example. 
(In fact, the torus-remark works only if we assume that each "region" in the torus is homeomorphic to a $2$-ball. This is true if the triangulation is fine enough, but it is not satisfied if you embedd a planar graph into a torus, for example. I wanted to include a torus-example to illustrate the gist of the topological approach, however.)
