I have the problem: Let $d>1$ be an integer. Prove that if every vertex of a graph $G$ has degree at least d, then G contains a cycle of length at least $d + 1$.
I'm pretty sure this can be done by induction and the base case would be a connected graph of 3 vertices for which every vertex has degree 2 and G has a cycle of length 3. I don't really know where to go from here though.
I think it may have something to do with the fact that the graph with minimum degree $d+1$ would need $d+1$ edges added to the graph which satisfies the proposition for $d$, but I don't know how to take that further.