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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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