Relating path coverings to vertex labellings with a condition at distance two Original Research Article

A λ-labelling of graph G is an integer labelling of V(G) such that adjacent vertices have labels that differ by at least two and vertices distance two apart have labels that differ by at least one. The λ number of G, λ(G), is the minimum span of labels over all such labellings. Griggs and Yeh have studied the relationship between λ(G) and graph invariants χ(G) and Δ (G). In this paper, we derive the relationship between λ(G) and another graph invariant, the path covering number of Gc. Applications include the determination of the λ-number of the join of two graphs, the product of two complete graphs, and the complete multi-partite graphs.