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arXiv · 1508.06112

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

Hervé Hocquard·Jakub Przybyło
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Abstract

A proper edge $k$-colouring of a graph $G=(V,E)$ is an assignment $c:E\to \{1,2,\ldots,k\}$ 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 $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$. We denote by $χ'_{\sum}(G)$ 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 $χ'_Σ(G) \leq Δ(G)+1$ for any graph $G$ without isolated edges and with ${\rm mad}(G)<3$, $Δ(G) \geq 6$.

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Hervé Hocquard, Jakub Przybyło. 2015-08-25. On the neighbour sum distinguishing index of graphs with bounded maximum average degree. https://doi.org/10.1007/s00373-017-1822-3

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