Abstract
A total weighting of a graph G is a mapping ϕ that assigns to each element z ∈ V (G)∪E(G) a weight ϕ(z). A total weighting ϕ is proper if for any two adjacent vertices u and v, ∑ e∈E(u) ϕ(e)+ϕ(u)≠∑ e∈E(v) ϕ(e)+ϕ(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 ϕ with ϕ(z) ∈ L(z) for each element z ∈ V (G)∪E(G).
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Grant numbers: NSC102-2115-M-110-006-MY2.
Grant numbers: NSF11171310 and ZJNSF Z6110786.