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Neighbour sum distinguishing total colourings via the combinatorial nullstellensatz. (English) Zbl 1330.05074
Summary: Consider a simple graph $$G=(V,E)$$ and its (proper) total colouring $$c$$ with elements of the set $$\{1,2,\dots,k\}$$. We say that $$c$$ is neighbour sum distinguishing if for every edge $$uv\in E$$, the sums of colours met by $$u$$ and $$v$$ differ, i.e., $$c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$$. The least $$k$$ guaranteeing the existence of such a colouring is denoted $$\chi^{\prime\prime}_\Sigma(G)$$. We investigate a daring conjecture presuming that $$\chi^{\prime\prime}_\Sigma(G)\leq \Delta(G)+3$$ for every graph $$G$$, a seemingly demanding problem if confronted with up-to-date progress in research on the total colouring conjecture itself. We note that $$\chi^{\prime\prime}_\Sigma(G)\leq \delta(G)+2\mathrm{col}(G)-1$$ and apply combinatorial nullstellensatz to prove a stronger bound: $$\chi^{\prime\prime}_\Sigma(G)\leq \delta (G)+\lceil 53\mathrm{col}(G)\rceil$$. This imply an upper bound of the form $$\chi^{\prime\prime}_\Sigma(G)\leq \delta (G)+\mathrm{const.}$$ for many classes of graphs with unbounded maximum degree. In particular we obtain $$\chi^{\prime\prime}_\Sigma(G)\leq \delta (G)+10$$ for planar graphs. In fact we show that identical bounds also hold if we use any set of $$k$$ real numbers instead of $$\{1,2,\dots,k\}$$ as edge colours, and moreover the same is true in list versions of the both concepts.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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