On the total neighbour sum distinguishing index of graphs with bounded maximum average degree

摘要

A proper total k-colouring of a graph G = (V, E) is an assignment c : V boolean OR E -> {1, 2, ..., k} of colours to the edges and the vertices of G such that no two adjacent edges or vertices and no edge and its end-vertices are associated with the same colour. A total neighbour sum distinguishing k-colouring, or tnsd k-colouring for short, is a proper total k-colouring such that Sigma(e(sic)u) c(e) + c(u) not equal Sigma(e(sic)v) c(e) + c(v) for every edge uv of G. We denote by chi(Sigma)" (G) the total neighbour sum distinguishing index of G, which is the least integer k such that a tnsd k-colouring of G exists. It has been conjectured that. chi(Sigma)" (G) = 8.