Altitude of \( K_{3,n}\)

An edge-ordering of a graph \( G = (V, E) \) is a one-to-one function \( f \) from \( E \) to the set of positive integers. A path of length \( k \) in \( G \) is called a \( (k, f) \)-ascent if \( f \) increases along the edge sequence of the path. The altitude \( \alpha(G) \) of \( G \) is the greatest integer \( k \) such that for all edge-orderings \( f \), \( G \) has a \( (k, f) \)-ascent.

We obtain a recursive lower bound for \( \alpha(K_{m,n}) \) and show that

\[\alpha(K_{3,n}) = \begin{cases}4 & \text{if } 5 \leq n \leq 9 \\5 & \text{if } 10 \leq n \leq 12 \\6 & \text{if } n \geq 13\end{cases}\]