On the Neighbour Sum Distinguishing Index of Graphs with Bounded Maximum Average Degree

A proper edge k -colouring of a graph G = ( V , E ) is an assignment c : E → { 1 , 2 , … , k } of colours to the edges of the graph such that no two adjacent edges are associated with the same colour. A neighbour sum distinguishing edge k -colouring, or nsd k -colouring for short, is a proper edge k -colouring such that ∑ e ∋ u c ( e ) ≠ ∑ e ∋ v c ( e ) for every edge uv of G . We denote by χ Σ ′ ( G ) the neighbour sum distinguishing index of G , which is the least integer k such that an nsd k -colouring of G exists. By definition at least maximum degree, Δ ( G ) colours are needed for this goal. In this paper we prove that χ Σ ′ ( G ) ≤ Δ ( G ) + 1 for any graph G without isolated edges, with mad ( G ) < 3 and Δ ( G ) ≥ 6 .