<p><b>Abstract</b>—Interval routing is a space-efficient method for point-to-point networks. It is based on labeling the edges of a network with intervals of vertex numbers (called interval labels). An <tmath>$M$</tmath>-label scheme allows up to <tmath>$M$</tmath> labels to be attached on an edge. For arbitrary graphs of size <tmath>$n$</tmath>, <tmath>$n$</tmath> the number of vertices, the problem is to determine the minimum <tmath>$M$</tmath> necessary for achieving optimality in the length of the longest routing path. The longest routing path resulted from a labeling is an important indicator of the performance of any algorithm that runs on the network. We prove that there exists a graph with <tmath>$D=\Omega(n^{\frac{1}{3}})$</tmath> such that if <tmath>$M\leq {\frac{n}{18D}}-O(\sqrt{\frac{n}{D}})$</tmath>, the longest path is no shorter than <tmath>$D+\Theta({\frac{D}{\sqrt{M}}})$</tmath>. As a result, for any <tmath>$M$</tmath>-label IRS, if the longest path is to be shorter than <tmath>$D+\Theta({\frac{D}{\sqrt{M}}})$</tmath>, at least <tmath>$M=\Omega({\frac{n}{D}})$</tmath> labels per edge would be necessary.</p>