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Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10. (English) Zbl 1426.05051
Summary: Given a simple graph \(G\), a proper total-\(k\)-coloring \(\phi : V(G) \cup E(G) \rightarrow \{1, 2, \ldots, k \}\) is called neighbor sum distinguishing if \(S_{\phi}(u) \neq S_{\phi}(v)\) for any two adjacent vertices \(u,v \in V(G)\), where \(S_{\phi}(u)\) is the sum of the color of \(u\) and the colors of the edges incident with \(u\). It has been conjectured by M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] that \(\mathop{\Delta}(G) + 3\) colors enable the existence of a neighbor sum distinguishing total coloring. The conjecture is confirmed for any graph with maximum degree at most 3 and for planar graph with maximum degree at least 11. We prove that the conjecture holds for any planar graph \(G\) with \(\mathop{\Delta}(G) = 10\). Moreover, for any planar graph \(G\) with \(\mathop{\Delta}(G) \geq 11, \mathop{\Delta}(G) + 2\) colors guarantee such a total coloring, and the upper bound \(\mathop{\Delta}(G) + 2\) is tight.

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