Optimal Edge-Numbering of Binary Trees

Abstract

Let G = (V,E) be a graph with vertex-set V(G) and edge-set E(G). A 1-1 mapping f: E(G)→{1,2,..., |E(G)|} will be called an edge-numbering of G. E(v) means the set of all edges of G which are incident to the vertex v of G. Define

$$ \begin{gathered} {{D}_{f}}(v) = \max \quad \left| {f(e) - f(e')} \right|\;and \hfill \\ \quad \quad \quad \;e,e' \in E(v) \hfill \\ {{W}_{f}}(G) = \max \quad {{D}_{f}}(v)\; \cdot \hfill \\ \quad \quad \quad \;v \in V(G) \hfill \\ \end{gathered} $$

$${{W}_{f}}\left( G \right)=\max {{D}_{f}}\left( v\right).$$