Prove that if $u$ and $v$ do not lie on a common cycle then $od(u)≠od(v)$

Question

Let $u$ and $v$ be 2 vertices in a tournament $T$. Prove that if $u$ and $v$ do not lie on a common cycle then $od(u)≠od(v)$

I have no idea how to start this proof. Please help.

Answer 1

Hint: Prove this by contrapositive

• Assume that $od(u)=od(v)$

• show that there exist $u-v$ path and $v-u$ path

• Conclude, $u,v$ lie on same cycle.

Let me know if there is any part you confused or unable to show.