Altitude of $K_{3,n}$

Abstract

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

$$\alpha (K_{3,n})=\{\begin{array}{ll}4& \text{if }5\le n\le 9\\ 5& \text{if }10\le n\le 12\\ 6& \text{if }n\ge 13\end{array}$$