A Relaxed Case on 1-2-3 Conjecture

In 2004, Karoński, ?uczak and Thomason posed the conjecture that every graph admits a vertex-coloring edge 3-weighting using colors 1, 2, 3. In this paper, we prove that this conjecture is true if the vertex-coloring is relaxed to be a tree-coloring. Furthermore, we verify that some classes of graphs permit tree-coloring edge 2-weightings.