On k-Absolute Difference Labelling of Graphs

A mapping which maps some set of graph elements to a set of numbers is said to be graph labelling. In this paper, a new variation of labelling called k-absolute difference labelling is introduced. A one-one mapping f : V → {0, 1, 2, ..., r} is called k- absolute difference vertex labelling of a graph G = (V, E), if |f (x)−f (y)| ≥ k for all (x, y) ∈ E. The smallest r for which a graph G has a k-absolute difference vertex labelling is called the k-absolute difference vertex strength of G and is denoted by AVS_k(G). A one- one mapping f : E → {0, 1, 2, ..., r} is called k- absolute difference edge labelling if  |f (e₁)−f (e₂)| ≥ k for all e₁, e₂ ∈ E and e₁, e₂ are adjacent.  The smallest r for which a graph G has a k-absolute difference edge labelling is called the k-absolute difference edge strength of G and is denoted by AES_k(G). Here, two new parameters AVS_k and AES_k of some special graphs are determined and bounds for general graph is presented.