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Neighbor sum distinguishing total colorings of \(K_4\)-minor free graphs. (English) Zbl 1306.05066
Summary: A total \([k]\)-coloring of a graph \(G\) is a mapping \(\phi :V(G)\cup E(G) \to\{1, 2, \dots, k\}\) such that any two adjacent elements in \(V(G)\cup E(G)\) receive different colors. Let \(f(v)\) denote the sum of the colors of a vertex \(v\) and the colors of all incident edges of \(v\). A total \([k]\)-neighbor sum distinguishing-coloring of \(G\) is a total \([k]\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(f(u)\neq f(v)\). By \(\chi_{\mathrm{nsd}}^{\prime\prime}(G)\), we denote the smallest value \(k\) in such a coloring of G. Pilśniak and Woźniak conjectured \(\chi_{\mathrm{nsd}}^{\prime\prime} (G)\Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for \(K_4\)-minor free graphs. Furthermore, we show that if \(G\) is a \(K_4\)-minor free graph with \(\Delta(G)\geqslant 4\), then \(\chi_{\mathrm{nsd}}^{\prime\prime}(G)\leqslant \Delta (G)+2\). The bound \(\Delta (G)+2\) is sharp.

MSC:
05C15 Coloring of graphs and hypergraphs
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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