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On the neighbor sum distinguishing total coloring of planar graphs. (English) Zbl 1331.05084
Summary: Let \(c\) be a proper total coloring of a graph \(G = (V, E)\) with integers \(1, 2, \ldots, k\). For any vertex \(v \in V(G)\), let \(\sum_c(v)\) denote the sum of colors of the edges incident with \(v\) and the color of \(v\). If for each edge \(uv \in E(G)\), \(\sum_c(u) \neq \sum_c(v)\), then such a total coloring is said to be neighbor sum distinguishing. The least \(k\) for which such a coloring of \(G\) exists is called the neighbor sum distinguishing total chromatic number and denoted by \(\chi_{\Sigma}^{\prime\prime}(G)\). M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured \(\chi_{\Sigma}^{\prime\prime}(G) \leq \Delta(G) + 3\) for any simple graph with maximum degree \(\Delta(G)\). It is known that this conjecture holds for any planar graph with \(\Delta(G) \geq 13\). In this paper, we prove that for any planar graph, \(\chi_{\Sigma}^{\prime\prime}(G) \leq \max \{\Delta(G) + 3, 14 \}\).

MSC:
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
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[1] Alon, N., Combinatorial nullstellensatz, Combin. Probab. Comput., 8, 7-29, (1999) · Zbl 0920.05026
[2] Chartrand, G.; Jacobson, M.; Lehel, J.; Oellermann, O.; Ruiz, S.; Saba, F., Irregular networks, Congr. Numer., 64, 197-210, (1988)
[3] Chen, X., On the adjacent vertex distinguishing total coloring numbers of graphs with \(\operatorname{\Delta} = 3\), Discrete Math., 308, 17, 4003-4007, (2008) · Zbl 1203.05052
[4] Cheng, X.; Huang, D.; Wang, G.; Wu, J., Neighbor sum distinguishing total colorings of planar graphs with maximum degree δ, Discrete Appl. Math., 190, 34-41, (2015) · Zbl 1316.05041
[5] Coker, T.; Johannson, K., The adjacent vertex distinguishing total chromatic number, Discrete Math., 312, 741-2750, (2012) · Zbl 1245.05042
[6] Ding, L.; Wang, G.; Yan, G., Neighbor sum distinguishing total colorings via the combinatorial nullstellensatz, Sci. Sin. Math., 57, 9, 1875-1882, (2014) · Zbl 1303.05058
[7] L. Ding, G. Wang, J. Wu, J. Yu, Neighbor sum (set) distinguishing total choosability via the Combinatorial Nullstellensatz, submitted for publication. · Zbl 1371.05078
[8] Dong, A.; Wang, G., Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree, Acta Math. Sinica, 30, 4, 703-709, (2014) · Zbl 1408.05061
[9] Huang, D.; Wang, W., Adjacent vertex distinguishing total coloring of planar graphs with large maximum degree, Sci. Sin. Math., 42, 2, 151-164, (2012), (in Chinese)
[10] Huang, D.; Wang, W.; Yan, C., A note on the adjacent vertex distinguishing total chromatic number of graphs, Discrete Math., 312, 24, 3544-3546, (2012) · Zbl 1258.05037
[11] Huang, P.; Wong, T.; Zhu, X., Weighted-1-antimagic graphs of prime power order, Discrete Math., 312, 14, 2162-2169, (2012) · Zbl 1244.05186
[12] Kalkowski, M.; Karoński, M.; Pfender, F., Vertex-coloring edge-weightings: towards the 1-2-3-conjecture, J. Combin. Theory Ser. B, 100, 347-349, (2010) · Zbl 1209.05087
[13] Karoński, M.; Łuczak, T.; Thomason, A., Edge weights and vertex colors, J. Combin. Theory Ser. B, 91, 1, 151-157, (2004) · Zbl 1042.05045
[14] Li, H.; Ding, L.; Liu, B.; Wang, G., Neighbor sum distinguishing total colorings of planar graphs, J. Comb. Optim., (2013)
[15] Li, H.; Liu, B.; Wang, G., Neighbor sum distinguishing total colorings of \(K_4\)-minor free graphs, Front. Math. China, 8, 6, 1351-1366, (2013) · Zbl 1306.05066
[16] Loeb, S.; Tang, Y., Asymptotically optimal neighbor sum distinguishing total colorings of graphs, (2015)
[17] Pilśniak, M.; Woźniak, M., On the total-neighbor-distinguishing index by sums, Graphs Combin., 31, 1-12, (2013)
[18] Przybyło, J., Irregularity strength of regular graphs, Electron. J. Combin., 15, 1, (2008) · Zbl 1163.05329
[19] Przybyło, J., Linear bound on the irregularity strength and the total vertex irregularity strength of graphs, SIAM J. Discrete Math., 23, 1, 511-516, (2009) · Zbl 1216.05135
[20] Przybyło, J.; Woźniak, M., On a 1, 2 conjecture, Discrete Math. Theor. Comput. Sci., 12, 1, 101-108, (2010) · Zbl 1250.05093
[21] Przybyło, J.; Woźniak, M., Total weight choosability of graphs, Electron. J. Combin., 18, (2011) · Zbl 1217.05202
[22] Scheim, E., The number of edge 3-colorings of a planar cubic graph as a permanent, Discrete Math., 8, 377-382, (1974) · Zbl 0281.05103
[23] Wang, W.; Huang, D., The adjacent vertex distinguishing total coloring of planar graphs, J. Comb. Optim., 27, 2, 379-396, (2014) · Zbl 1319.90076
[24] Wang, W.; Wang, P., On adjacent-vertex-distinguishing total coloring of \(K_4\)-minor free graphs, Sci. China Ser. A, 39, 12, 1462-1472, (2009)
[25] Wang, Y.; Wang, W., Adjacent vertex distinguishing total colorings of outerplanar graphs, J. Comb. Optim., 19, 123-133, (2010) · Zbl 1216.05039
[26] Wong, T.; Zhu, X., Total weight choosability of graphs, J. Graph Theory, 66, 198-212, (2011) · Zbl 1228.05161
[27] Wong, T.; Zhu, X., Antimagic labelling of vertex weighted graphs, J. Graph Theory, 70, 3, 348-359, (2012) · Zbl 1244.05192
[28] Zhang, Z.; Chen, X.; Li, J.; Yao, B.; Lu, X.; Wang, J., On adjacent-vertex-distinguishing total coloring of graphs, Sci. China Ser. A, 48, 3, 289-299, (2005) · Zbl 1080.05036
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