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On the total neighbour sum distinguishing index of graphs with bounded maximum average degree
A proper total -colouring of a graph is an assignment of colours to the edges and the vertices of
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
-colouring, or tnsd -colouring for short, is a proper total -colouring
such that for every edge
of . We denote by the total neighbour sum
distinguishing index of , which is the least integer such that a tnsd
edge -colouring of exists. It has been conjectured that
for every graph . In this paper we
confirm this conjecture for any graph with and
.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1508.0611