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Neighbor sum distinguishing total coloring of IC-planar graphs. (English) Zbl 1445.05043
The authors consider the problem of neighbor sum distinguishing total coloring of IC-planar graphs. A proper total-$$k$$-coloring of a graph $$G$$ is a mapping $$c: V(G)\cup E(G) \rightarrow \{1, 2, \ldots, k\}$$ such that any two adjacent elements in $$V(G) \cup E(G)$$ receive different colors. Let $$\sum_c(u)$$ denote the sum of the color of a vertex $$v$$ and the colors of all edges incident with $$v$$. If for each edge $$uv \in E(G)$$, $$\sum_c(u) \ne\sum_c (v)$$, then such a proper total-$$k$$-coloring is called a $$k$$-neighbor sum distinguishing total coloring, denoted by tnsd-$$k$$-coloring, for short. The least number $$k$$ needed for such a coloring of $$G$$, denoted by $$\chi^{\prime\prime}_{\Sigma}(G)$$, is the neighbor sum distinguishing total chromatic number. The problem is very well known in the literature. There is a known conjecture, posed by M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)], that $$\chi^{\prime\prime}_{\Sigma}(G)\leq \Delta(G)+3$$ for any graph $$G$$. Also some other bounds on $$\chi^{\prime\prime}_{\Sigma}(G)$$are known proved for general or particular classes of graphs.
This paper contributes a lot to the knowledge in the area of neighbor sum distinguishing total coloring by proving that $$\chi^{\prime\prime}_{\Sigma}(G)\leq \Delta(G)+2$$ for IC-planar graphs without 2-vertex incident with crossed edge and with $$\Delta(G)\geq 14$$. The proof of the result is very interesting and carried with great care.
I find the paper very valuable.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C07 Vertex degrees 05C35 Extremal problems in graph theory
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