Proof of cycles of different length

Question

Let $G$ be a graph with $\delta(G)\geq k$. Prove that $G$ contains at least $k-1$ cycles such that no two of which have the same length.

Any proofs or hints are greatly appreciated, I am lost on how to prove this.

Answer 1

The statement is not true if loops or multiple edges are allowed. So, let $G$ be a graph with minimum degree at least $k$, let $P$ be a maximal path and $v$ an endpoint of $P$. At least $k-1$ edges different from the last edge of $P$ emerge from $v$. They must all connect to one of the other vertices on $P$, otherwise we contradict the maximality of $P$.

Because $G$ is simple, these edges must all connect to different vertices of $P$, and this exposes at least $k-1$ cycles with different lengths.