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

H. Hocquard () and J. Przybyło ()
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H. Hocquard: University of Bordeaux
J. Przybyło: AGH University of Science and Technology

Journal of Combinatorial Optimization, 2020, vol. 39, issue 2, No 7, 412-424

Abstract: Abstract A proper total k-colouring of a graph $$G=(V,E)$$G=(V,E) is an assignment $$c : V \cup E\rightarrow \{1,2,\ldots ,k\}$$c:V∪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 $$\sum _{e\ni u}c(e)+c(u)\ne \sum _{e\ni v}c(e)+c(v)$$∑e∋uc(e)+c(u)≠∑e∋vc(e)+c(v) for every edge uv of G. We denote by $$\chi ''_{\Sigma }(G)$$χΣ′′(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 $$\chi ''_{\Sigma }(G) \le \Delta (G) + 3$$χΣ′′(G)≤Δ(G)+3 for every graph G. In this paper we confirm this conjecture for any graph G with $$\mathrm{mad}(G)

Keywords: Total neighbour sum distinguishing index; Maximum average degree; Combinatorial Nullstellensatz; Discharging method (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-019-00480-4

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