Graph theory - perfect matching using Petersen's theorem

HI guys ,

I have a little problem regarding Perfect Matching .

I have a graph G=(V,E) with |V|=2n and δ(G)>=n (where δ(G) is the minimal degree

for each vertex) .

Now I need to prove that G has a perfect matching .

I tried to use Perersen's theorem for 3-regular graph but I kinda got lost (Petersen graph - Wikipedia, the free encyclopedia

can I say that if I separate the 2n vertexes that I have to 2 groups , S and T ,

then that the worst case scenario would be that : S would have at least two

neighbors in T , and further more , that there would be NO bridges ?

I cannot understand why , if we take some vertex , in would have a least two neighbors ??!!

if so , then now I want to use Petersen's theorem : if there are no bridges , is it right to say that any n-regular graph would have a 3-regular graph ?

I'll appreciate your help

thanks (: