Relaxed Graceful Labellings of Trees

A graph G on m edges is considered graceful if there is a labelling f of the vertices of G with distinct integers in the set {0, 1, . . . ,m} such that the induced edge labelling g defined by g(uv) = |f(u) − f(v)| is a bijection to {1, . . . ,m}. We here consider some relaxations of these conditions as applied to tree labellings: 1. Edge-relaxed graceful labellings, in which repeated edge labels are allowed, 2. Range-relaxed graceful labellings, in which the upper bound m′ is allowed to go higher than the number of edges, and 3. Vertex-relaxed graceful labellings, in which repeated vertex labels are allowed. The first of these had been looked at by Rosa and Širáň [10]. Here some linear bounds in the relevant metrics are given for range-relaxed and vertex-relaxed graceful labellings.