Let H be a cycle with an extra chord. Let $(A,B)$ be a nontrival partition of $V(H)$. Then unless H is bipartite between $A$ and $B$, $H$ contains paths of every length $l$ $\lt$ $|V(H)|$ which begin in $A$ and end in $B$.
I looked for the proof which used gcd and modular arithmetic and followed it but then got lost somewhere in middle of it and now I am not being able to move ahead. Can anyone provide a reference to a different and easier proof to this result. Here is a link to what I read https://www.dpmms.cam.ac.uk/~dc340/EGT10.pdf