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

Author

Listed:
  • H. Hocquard

    (University of Bordeaux)

  • J. Przybyło

    (AGH University of Science and Technology)

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)

Suggested Citation

  • H. Hocquard & J. Przybyło, 2020. "On the total neighbour sum distinguishing index of graphs with bounded maximum average degree," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 412-424, February.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:2:d:10.1007_s10878-019-00480-4
    DOI: 10.1007/s10878-019-00480-4
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    References listed on IDEAS

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    1. Hualong Li & Laihao Ding & Bingqiang Liu & Guanghui Wang, 2015. "Neighbor sum distinguishing total colorings of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 675-688, October.
    2. Yang, Donglei & Sun, Lin & Yu, Xiaowei & Wu, Jianliang & Zhou, Shan, 2017. "Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 456-468.
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