Geodesic
Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 243-253.

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For a given graph G = (V (G), E(G)), a proper total coloring ϕ : V (G) ∪ E(G) →1, 2, . . ., k is neighbor sum distinguishing if f(u) f(v) for each edge uv ∈ E(G), where f(v) = Σ_ uv ∈ E(G) ϕ (uv) + ϕ (v), v ∈ V (G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χ_Σ^” (G). Pilśniak and Woźniak first introduced this coloring and conjectured that χ_Σ^”(G) ≤Δ (G)+3 for any graph with maximum degree Δ (G). In this paper, by using the discharging method, we prove that for any planar graph G without 5-cycles, χ_Σ^” (G) ≤max{Δ (G)+2, 10 }. The bound Δ (G) + 2 is sharp. Furthermore, we get the exact value of χ_Σ^” (G) if Δ (G) ≥ 9.
Keywords: neighbor sum distinguishing total coloring, discharging method, planar graph 
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Zhao, Xue; Xu, Chang-Qing. Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 243-253. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a15/

[1] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (North-Holland, NewYork-Amsterdam-Oxford, 1982).

[2] X. Cheng, D. Huang, G. Wang and J. Wu, Neighbor sum distinguishing total colorings of planar graphs with maximum degree Δ, Discrete Appl. Math. 190 – 191 (2015) 34–41. doi:10.1016/j.dam.2015.03.013

[3] A. Dong and G. Wang, Neighbor sum distinguising 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

[4] L. Ding, G. Wang and G. Yan, Neighbor sum distinguising total colorings via the Combinatorial Nullstellensatz, Sci. China Math. 57 (2014) 1875–1882. doi:10.1007/s11425-014-4796-0

[5] S. Ge, J. Li and C. Xu, Neighbor sum distinguishing total chromatic number of planar graphs without 4 -cycles, Util. Math. 105 (2017) 259–265.

[6] S. Ge, J. Li and C. Xu, Neighbor sum distinguishing total coloring of planar graphs without 5 -cycles, Theoret. Comput. Sci. 689 (2017) 169–175. doi:10.1016/j.tcs.2017.05.037

[7] 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

[8] J. Li, S. Ge and C. Xu, Neighbor sum distinguishing total colorings of planar graphs with girth at least 5, Util. Math. 104 (2017) 115–121.

[9] H. Li, B. Liu and G. Wang, Neighbor sum distinguishing total colorings of K4-minor free graphs, Front. Math. China 8 (2013) 1351–1366. doi:10.1007/s11464-013-0322-x

[10] Q. Ma, J. Wang and H. Zhao, Neighbor sum distinguishing total colorings of planar graphs without short cycles, Util. Math. 98 (2015) 349–359.

[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

[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

[14] H. Song, W. Pan, X. Gong and C. Xu, A note on the neighbor sum distinguishing total coloring of planar graphs, Theoret. Comput. Sci. 640 (2016) 125–129. doi:10.1016/j.tcs.2016.06.007

[15] H. Song and C. Xu, Neighbor sum distinguishing total chromatic number of K4-minor free graph, Front. Math. China 12 (2017) 937–947. doi:10.1007/s11464-017-0649-9

[16] H. Song and C. Xu, Neighbor sum distinguishing total coloring of planar graphs without 4 -cycles, J. Comb. Optim. 34 (2017) 1147–1158. doi:10.1007/s10878-017-0137-x

[17] 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

[18] J. Wang, J. Cai and B. Qiu, Neighbor sum distinguishing total choosability of planar graphs without adjacent triangles, Theoret. Comput. Sci. 661 (2017) 1–7. doi:10.1016/j.tcs.2016.11.003

[19] J. Wang, Q. Ma and X. Han, Neighbor sum distinguishing total colorings of triangle free planar graphs, Acta Math. Sin. (Engl. Ser.) 31 (2015) 216–224. doi:10.1007/s10114-015-4114-y

[20] D. Yang, X. Yu, L. Sun, J. Wu and S. Zhou, Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10, Appl. Math. Comput. 314 (2017) 456–468. doi:10.1016/j.amc.2017.06.002

[21] J. Yao, X. Yu, G. Wang and C. Xu, Neighbor sum distinguishing total coloring of 2 -degenerate graphs, J. Comb. Optim. 34 (2017) 64–70. doi:10.1007/s10878-016-0053-5

[22] 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