Abstract
A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2, ..., k}. Let f(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E(G), f(u) ≠ f(v). By χ″nsd(G), we denote the smallest value k in such a coloring of G. Pilśniak and Woźniak conjectured that χ″nsd(G) ≤ Δ(G)+3 for any simple graph with maximum degree Δ(G). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.
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References
Alon, N.: Combinatorial Nullstellensatz. Combin. Probab. Comput., 8, 7–29 (1999)
Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications, North-Holland, New York, 1976
Chen, X.: On the adjacent vertex distinguishing total coloring numbers of graphs with Δ = 3. Discrete Math., 308(17), 4003–4007 (2008)
Ding, L., Wang, G., Yan, G.: Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz. Sci. China Ser. A, 57(9), 1875–1882 (2014)
Ding, L., Wang, G., Wu, J., Yu, J.: Neighbor sum (set) distinguishing total choosability via the Combinatorial Nullstellensatz, submitted
Dong, A., Wang, G.: Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree. Acta Math. Sin., Engl. Series, 30(4), 703–709 (2014)
Huang, D., Wang, W.: Adjacent vertex distinguishing total coloring of planar graphs with large maximum degree (in Chinese). Sci. Sin. Math., 42(2), 151–164 (2012)
Huang, P., Wong, T., Zhu, X.: Weighted-1-antimagic graphs of prime power order. Discrete Math., 312(14), 2162–2169 (2012)
Karoński, M., Łuczak, T., Thomason, A.: Edge weights and vertex colours. J. Combin. Theory Ser. B, 91(1), 151–157 (2004)
Li, H., Ding, L., Liu, B., et al.: Neighbor sum distinguishing total colorings of planar graphs. J. Comb. Optim., DOI: 10.1007/s10878-013-9660-6
Li, H., Liu, B., Wang, G.: Neighor sum distinguishing total colorings of K 4-minor free graphs. Front. Math. China, 8(6), 1351–1366 (2013)
Pilśniak, M., Woźniak, M.: On the total-neighbor-distinguishing index by sums. Graph and Combin., DOI 10.1007/s00373-013-1399-4
Przybyło, J.: Irregularity strength of regular graphs. Electron. J. Combin., 15(1), #R82, 10pp (2008)
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)
Przybyło, J., Woźniak, M.: On a 1, 2 conjecture. Discrete Math. Theor. Comput. Sci., 12(1), 101–108 (2010)
Przybyło, J., Woźniak, M.: Total weight choosability of graphs. Electron. J. Combin., 18, #P112, 11pp (2011)
Seamone, B.: The 1-2-3 conjecture and related problems: a survey, arXiv:1211.5122
Wang, W., Huang, D.: The adjacent vertex distinguishing total coloring of planar graphs. J. Comb. Optim., DOI 10.1007/s10878-012-9527-2
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)
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)
Wong, T., Zhu, X.: Total weight choosability of graphs. J. Graph Theory, 66, 198–212 (2011)
Wong, T., Zhu, X.: Antimagic labelling of vertex weighted graphs. J. Graph Theory, 3(70), 348–359 (2012)
Zhang, Z., Chen, X., Li, J., et al.: On adjacent-vertex-distinguishing total coloring of graphs. Sci. China, Ser. A, 48(3), 289–299 (2005)
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Supported by National Natural Science Foundation of China (Grant No. 11201180) and the Scientific Research Foundation of University of Ji’nan (Grant No. XKY1120)
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Wang, J.H., Ma, Q.L. & Han, X. Neighbor sum distinguishing total colorings of triangle free planar graphs. Acta. Math. Sin.-English Ser. 31, 216–224 (2015). https://doi.org/10.1007/s10114-015-4114-y
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DOI: https://doi.org/10.1007/s10114-015-4114-y