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Neighbor sum distinguishing total coloring of planar graphs without 4-cycles. (English) Zbl 1378.05073
Summary: Let $$G=(V,E)$$ be a graph and $$\phi:V\cup E\rightarrow \{1,2,\dots,k\}$$ be a proper total coloring of $$G$$. Let $$f(v)$$ denote the sum of the color on a vertex $$v$$ and the colors on all the edges incident with $$v$$. The coloring $$\phi$$ is neighbor sum distinguishing if $$f(u)\neq f(v)$$ for each edge $$uv\in E(G)$$. The smallest integer $$k$$ in such a coloring of $$G$$ is the neighbor sum distinguishing total chromatic number of $$G$$, denoted by $$\chi^{\prime\prime}_\Sigma(G)$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured that $$\chi^{\prime\prime}_\Sigma(G)\leq \Delta (G)+3$$ for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that $$\chi^{\prime\prime}_\Sigma(G)\leq\max\{\Delta (G)+2,10\}$$ for planar graph $$G$$ without 4-cycles. The bound $$\Delta (G)+2$$ is sharp if $$\Delta (G)\geq 8$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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##### References:
 [1] Alon, N, Combinatorial nullstellensatz, Comb Probab Comput, 8, 7-29, (1999) · Zbl 0920.05026 [2] Bondy J, Murty U (1976) Graph theory with applications. North-Holland, New York · Zbl 1226.05083 [3] 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 [4] Ding L, Wang G, Wu J, Yu J (2014) Neighbor sum (set) distinguising total choosability via the Combinatorial Nullstellensatz (submitted) · Zbl 1345.05035 [5] Ding, L; Wang, G; Yan, G, Neighbour sum distinguishing total colorings via the combinatorial nullstellensatz, Sci China Math, 57, 1875-1882, (2014) · Zbl 1303.05058 [6] Li, H; Ding, L; Liu, B; Wang, G, Neighbor sum distinguishing total colorings of planar graphs, J Comb Optim, 30, 675-688, (2015) · Zbl 1325.05083 [7] Li, H; Liu, B; Wang, G, Neighbor sum distinguishing total colorings of $$K_{4}$$-minor free graphs, Front Math China, 8, 1351-1366, (2013) · Zbl 1306.05066 [8] Pilśniak, M; Woźniak, M, On the adjacent-vertex-distinguishing index by sums in total proper colorings, Graphs Comb, (2013) · Zbl 1312.05054 [9] Przybyło, J, Neighbour sum distinguishing total colorings via the combinatorial nullstellensatz, Discrete Appl Math, 202, 163-173, (2016) · Zbl 1330.05074 [10] Qu, C; Wang, G; Wu, J; Yu, X, On the neighbour sum distinguishing total coloring of planar graphs, Theor Comput Sci, 609, 162-170, (2016) · Zbl 1331.05084 [11] Qu C, Wang G, Yan G, Yu X (2016) Neighbor sum distinguishing total choosability of planar graphs. J Comb Optim 32(3):906-916 · Zbl 1348.05082 [12] Song, H; Pan, W; Gong, X; Xu, C, A note on the neighbor sum distinguishing total coloring of planar graphs, Theor Comput Sci, 640, 125-129, (2016) · Zbl 1345.05035 [13] Wang, J; Cai, J; Ma, Q, Neighbor sum distinguishing total choosability of planar graphs without 4-cycles, Discrete Appl Math, 206, 215-219, (2016) · Zbl 1335.05051 [14] Wang G, Ding L, Cheng X, Wu J Improved bounds for neighobr sum (set) distinguishing choosability of planar graphs. SIAM Discrete Math (submitted) · Zbl 1342.05052 [15] Wang, J; Ma, Q; Han, X, Neighbor sum distinguishing total colorings of triangle free planar graphs, Acta Math Sin Engl Ser, 31, 216-224, (2015) · Zbl 1317.05065 [16] Wang J, Ma Q, Han X, Wang X (2016) A proper tatal coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles. J Comb Optim 32(2):626-638 · Zbl 1343.05066 [17] Yao, J; Yu, X; Wang, G; Xu, C, Neighbor sum distinguishing total coloring of 2-degenerate graphs, J Comb Optim, (2016) · Zbl 1342.05052 [18] Yao, J; Shao, Z; Xu, C, Neighbor sum distinguishing total choosability of graphs with $$Δ =3$$, Adv Math (China), 45, 343-348, (2016) · Zbl 1363.05087 [19] Yao, J; Yu, X; Wang, G; Xu, C, Neighbour sum (set) distinguishing total choosability of $$d$$-degenerate graphs, Graphs Comb, 32, 1611-1620, (2016) · Zbl 1342.05052
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