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Neighbor sum distinguishing total choosability of planar graphs without 4-cycles. (English) Zbl 1335.05051
Summary: Let $$G = (V, E)$$ be a graph and $$\phi$$ be a total $$k$$-coloring of $$G$$ by using the color set $$\{1, \ldots, k \}$$. Let $$\sum_\phi(u)$$ denote the sum of the color of the vertex $$u$$ and the colors of all incident edges of $$u$$. A $$k$$-neighbor sum distinguishing total coloring of $$G$$ is a total $$k$$-coloring of $$G$$ such that for each edge $$u v \in E(G)$$, $$\sum_\phi(u) \neq \sum_\phi(v)$$. By $$\chi_\Sigma^{\prime\prime}(G)$$, we denote the smallest value $$k$$ in such a coloring of $$G$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] first introduced this coloring and conjectured that $$\chi_\Sigma^{\prime\prime}(G) \leq \Delta(G) + 3$$ for any simple graph $$G$$. Let $$L_z(z \in V \cup E)$$ be a set of lists of integer numbers, each of size $$k$$. The smallest $$k$$ for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from $$L_z$$ for each $$z \in V \cup E$$ is called the neighbor sum distinguishing total choosability of $$G$$, and denoted by $$\mathrm{ch}_\Sigma^{\prime\prime}(G)$$. In this paper, we prove that $$\mathrm{ch}_\Sigma^{\prime\prime}(G) \leq \Delta(G) + 3$$ for planar graphs without 4-cycles with $$\Delta(G) \geq 7$$. This implies that M. Pilśniak and M. Woźniak’s conjecture [loc. cit.] is true for planar graphs without 4-cycles.

MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C15 Coloring of graphs and hypergraphs
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