On the Complexity of Gap-[2]-vertex-labellings of Subcubic Bipartite Graphs

A gap-[k]-vertex-labelling of a simple graph G = (V, E) is a pair (π, c_π) in which π : V (G) → {1, 2, ..., k} is an assignment of labels to the vertices of G and c_π : V (G) → {0, 1, ..., k} is a proper vertex-colouring of G such that, for every v ∈ V (G) of degree at least two, c_π(v) is induced by the largest difference, i.e. the largest gap, between the labels of its neighbours (cases where d(v) = 1 and d(v) = 0 are treated separately). Introduced in 2013 by A. Dehghan et al. [Dehghan, A., M. Sadeghi and A. Ahadi, Algorithmic complexity of proper labeling problems, Theoretical Computer Science 495 (2013), pp. 25–36.], they show that deciding whether a bipartite graph admits a gap-[2]-vertex-labelling is NP-complete and question the computational complexity of deciding whether cubic bipartite graphs admit such a labelling. In this work, we advance the study of the computational complexity for this class, proving that this problem remains NP-complete even when restricted to subcubic bipartite graphs.