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Neighbor sum distinguishing total choosability of IC-planar graphs. (English) Zbl 1430.05023
Summary: Two distinct crossings are independent if the end-vertices of the crossed pair of edges are mutually different. If a graph $$G$$ has a drawing in the plane such that every two crossings are independent, then we call $$G$$ a plane graph with independent crossings or IC-planar graph for short. A proper total-$$k$$-coloring of a graph $$G$$ is a mapping $$c : V (G) \cup E(G) \rightarrow \{1, 2, \dots, k\}$$ such that any two adjacent elements in $$V (G) \cup E(G)$$ receive different colors. Let $$\Sigma_c(v)$$ denote the sum of the color 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 $$uv \in E(G)$$, $$\Sigma_c(u) \neq \Sigma_c(v)$$. The least number $$k$$ needed for such a coloring of $$G$$ is the neighbor sum distinguishing total chromatic number, denoted by $$\chi^{\prime\prime}_\Sigma ( G )$$. In this paper, it is proved that if $$G$$ is an IC-planar graph with maximum degree $$\Delta (G)$$, then $$ch^{\prime\prime}_\Sigma( G ) \le \max \left\{\Delta(G ) + 3,17\right\}$$, where $$ch^{\prime\prime}_\Sigma(G)$$ is the neighbor sum distinguishing total choosability of $$G$$.

MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C62 Graph representations (geometric and intersection representations, etc.) 05C15 Coloring of graphs and hypergraphs 05C35 Extremal problems in graph theory 05C07 Vertex degrees
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References:
 [1] M.O. Albertson, Chromatic number, independence ratio, and crossing number, Ars Math. Contemp. 1 (2008) 1-6. · Zbl 1181.05032 [2] N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999) 7-29. doi:10.1017/S0963548398003411 [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (North-Holland, New York-Amsterdam-Oxford, 1982). · Zbl 1226.05083 [4] L. Ding, G. Wang and G. Yan, Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz, Sci. China Math. 57 (2014) 1875-1882. doi:10.1007/s11425-014-4796-0 · Zbl 1303.05058 [5] L. Ding, G. Wang, J. Wu and J. Yu, Neighbor sum (set) distinguishing total choosability via the Combinatorial Nullstellensatz, Graphs Combin. 33 (2017) 885-900. doi:10.1007/s00373-017-1806-3 · Zbl 1371.05078 [6] A. Dong and G. Wang, Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree, Acta Math. Sin. (Engl. Ser.) 30 (2014) 703-709. doi:10.1007/s10114-014-2454-7 · Zbl 1408.05061 [7] D. Král and L. Stacho, Coloring plane graphs with independent crossings, J. Graph Theory 64 (2010) 184-205. doi:10.1002/jgt.20448 · Zbl 1208.05019 [8] H. Li, B. Liu and G. Wang, Neighbor sum distinguishing total colorings of K_4-minor free graphs, Front. Math. China 8 (2013) 1351-1366. doi:10.1007/s11464-013-0322-x · Zbl 1306.05066 [9] H. Li, L. Ding, B. Liu and G. Wang, Neighbor sum distinguishing total colorings of planar graphs, J. Comb. Optim. 30 (2015) 675-688. doi:10.1007/s10878-013-9660-6 · Zbl 1325.05083 [10] S. Loeb, J. Przybyło and Y. Tang, Asymptotically optimal neighbor sum distinguishing total colorings of graphs, Discrete Math. 340 (2017) 58-62. doi:10.1016/j.disc.2016.08.012 · Zbl 1351.05083 [11] M. Pilśniak and M. Woźniak, On the total-neighbor-distinguishing index by sums, Graphs Combin. 31 (2015) 771-782. doi:10.1007/s00373-013-1399-4 [12] C. Qu, G. Wang, J. Wu and X. Yu, On the neighbor sum distinguishing total coloring of planar graphs, Theoret. Comput. Sci. 609 (2016) 162-170. doi:10.1016/j.tcs.2015.09.017 · Zbl 1331.05084 [13] C. Qu, G. Wang, G. Yan and X. Yu, Neighbor sum distinguishing total choosability of planar graphs, J. Comb. Optim. 32 (2016) 906-916. doi:10.1007/s10878-015-9911-9 · Zbl 1348.05082 [14] D.E. Scheim, The number of edge 3-colorings of a planar cubic graph as a permanent, Discrete Math. 8 (1974) 377-382. doi:10.1016/0012-365X(74)90157-5 · Zbl 0281.05103 [15] J. Wang, J. Cai and Q. Ma, Neighbor sum distinguishing total choosability of planar graphs without 4-cycles, Discrete Appl. Math. 206 (2016) 215-219. doi:10.1016/j.dam.2016.02.003 · Zbl 1335.05051 [16] J. Yao and H. Kong, Neighbor sum distinguishing total choosability of graphs with larger maximum average degree, Ars Combin. 125 (2016) 347-360. · Zbl 1413.05353 [17] X. Zhang and J. Wu, On edge colorings of 1-planar graphs, Inform. Process. Lett. 111 (2011) 124-128. doi:10.1016/j.ipl.2010.11.001 · Zbl 1259.05050 [18] J. Yao, X. Yu, G. Wang and C. Xu, Neighbor sum (set) distinguishing total choosability of d-degenerate graphs, Graphs Combin. 32 (2016) 1611-1620. doi:10.1007/s00373-015-1646-y · Zbl 1342.05052 [19] X. Cheng, D. Huang, G. Wang and J. Wu, Neighbor sum distinguishing total colorings of planar graphs with maximum degree, Discrete Appl. Math. 190-191 (2015) 34-41. doi:10.1016/j.dam.2015.03.013 · Zbl 1316.05041 [20] Y. Lu, M.M. Han and R. Luo, Neighbor sum distinguishing total coloring and list neighbor sum distinguishing total coloring, Discrete Appl. Math. 237 (2018) 109-115. doi:10.1016/j.dam.2017.12.001 · Zbl 1380.05076
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