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

  • H. Hocquard
  • J. PrzybyłoEmail author


A proper total k-colouring of a graph \(G=(V,E)\) is an assignment \(c : V \cup E\rightarrow \{1,2,\ldots ,k\}\) 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 \(\sum _{e\ni u}c(e)+c(u)\ne \sum _{e\ni v}c(e)+c(v)\) 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 \(\chi ''_{\Sigma }(G) \le \Delta (G) + 3\) for every graph G. In this paper we confirm this conjecture for any graph G with \(\mathrm{mad}(G)<\frac{14}{3}\) and \(\Delta (G) \ge 8\).


Total neighbour sum distinguishing index Maximum average degree Combinatorial Nullstellensatz Discharging method 



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Authors and Affiliations

  1. 1.CNRS, Bordeaux INP, LaBRI, UMR 5800University of BordeauxTalenceFrance
  2. 2.AGH University of Science and TechnologyKrakówPoland

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