Abstract
A total weighting of a graph G is a mapping ? that assigns to each element z ? V (G)?E(G) a weight ?(z). A total weighting ? is proper if for any two adjacent vertices u and v, ?e?E(u)?(e)+?(u)??e?E(v)?(e)+?(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ? with ?(z) ? L(z) for each element z ? V (G)?E(G).
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