Edge-Injective and Edge-Surjective Vertex Labellings

For a graph G = (V,E) we consider vertex-k-labellings f : V → {1, 2, . . . , k} for which the induced edge weighting w : E → {2, 3, . . . , 2k} with w(uv) = f(u) + f(v) is injective or surjective or both. We study the relation between these labellings and the number theoretic notions of an additive basis and a Sidon set, present a new construction for a so-called restricted additive basis and derive the corresponding consequences for the labellings. We prove that a tree of order n and maximum degree ∆ has a vertex-k-labelling f for which w is bijective if and only if ∆ ≤ k = n/2. Using this result we prove a recent conjecture of Ivančo and Jendrol’ concerning edge-irregular total labellings for graphs that are sparse enough.