Abstract
Given a simple graph G, an ordered pair (π, cπ) is said to be a gap- [k]-edge-labelling (resp. gap-[k]-vertex-labelling) ofG ifπ is an edge-labelling (vertex-labelling) on the set {1, . . . , k}, and cπ is a proper vertex-colouring such that every vertex of degree at least two has its colour induced by the largest difference among the labels of its incident edges (neighbours). The decision problems associated with these labellings are NP-complete for k ≥ 3, and even when k = 2 for some classes of graphs. This thesis presents a study of the computational complexity of these problems, structural properties for certain families of graphs and several labelling algorithms and techniques. First, we present an NP-completeness result for the family of subcubic bipartite graphs. Second, we present polynomial-time algorithms for families ofgraphs. Third, we introduce a new parameter associated with gap-[k]-vertex-labellings ofgraphs.
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