The undirected optical indices of trees

For a connected graph G, an instance I is a set of pairs of vertices and a corresponding routing R is a set of paths specified for all vertex-pairs in I. Let \(\mathfrak {R}_I\) be the collection of all routings with respect to I. The undirected optical index of G with respect to I refers to the minimum integer k to guarantee the existence of a mapping \(\phi :R\rightarrow \{1,2,\ldots ,k\}\), such that \(\phi (P)\ne \phi (P')\) if P and \(P'\) have common edge(s), over all routings \(R\in \mathfrak {R}_I\). A natural lower bound of the undirected optical index is the edge-forwarding index, which is defined to be the minimum of the maximum edge-load over all possible routings. Let w(G, I) and \(\pi (G,I)\) denote the undirected optical index and edge-forwarding index with respect to I, respectively. In this paper, we derive the inequality \(w(T,I_A)<\frac{3}{2}\pi (T,I_A)\) for any tree T, where \(I_A:=\{\{x,y\}:\,x,y\in V(T)\}\) is the all-to-all instance.