Hamiltonicity and circular distance two labellings

A k-circular distance two labelling (or k-c-labelling) of a graph G is a vertex-labelling such that the circular difference (mod k) of the labels is at least two for adjacent vertices, and at least one for vertices at distance two. Given G, denote σ(G) the minimum k for which there exists a k-c-labelling of G. Suppose G has n vertices, we prove σ(G)⩽n if Gc is Hamiltonian; and σ(G)=n+pv(Gc) otherwise, where pv(G) is the path covering number of G. We give exact values of σ(G) for some families of graphs such that Gc is Hamiltonian, and discuss injective k-c-labellings especially for joins and unions of graphs.