# zbMATH — the first resource for mathematics

Neighbor sum distinguishing total colorings of triangle free planar graphs. (English) Zbl 1317.05065
Summary: A total $$k$$-coloring $$c$$ of a graph $$G$$ is a proper total coloring $$c$$ of $$G$$ using colors of the set $$[k]=\{1,2,\dots ,k\}$$. 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 $$uv\in E(G)$$, $$f(u)\neq f(v)$$. By $$\chi''_{\mathrm{nsd}}(G)$$, we denote the smallest value $$k$$ in such a coloring of $$G$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured that $$\chi''_{\mathrm{nsd}}(G)\leq \Delta (G)+3$$ for any simple graph with maximum degree $$\Delta(G)$$. 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.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
Full Text:
##### References:
  Alon, N, Combinatorial nullstellensatz, Combin. Probab. Comput., 8, 7-29, (1999) · Zbl 0920.05026  Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications, North-Holland, New York, 1976 · Zbl 1226.05083  Chen, X, On the adjacent vertex distinguishing total coloring numbers of graphs with δ = 3, Discrete Math., 308, 4003-4007, (2008) · Zbl 1203.05052  Ding, L; Wang, G; Yan, G, Neighbor sum distinguishing total colorings via the combinatorial nullstellensatz, Sci. China Ser. A, 57, 1875-1882, (2014) · Zbl 1303.05058  Ding, L., Wang, G., Wu, J., Yu, J.: Neighbor sum (set) distinguishing total choosability via the Combinatorial Nullstellensatz, submitted · Zbl 1371.05078  Dong, A; Wang, G, Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree, Acta Math. Sin., Engl. Series, 30, 703-709, (2014) · Zbl 1408.05061  Huang, D; Wang, W, Adjacent vertex distinguishing total coloring of planar graphs with large maximum degree (in Chinese), Sci. Sin. Math., 42, 151-164, (2012)  Huang, P; Wong, T; Zhu, X, Weighted-1-antimagic graphs of prime power order, Discrete Math., 312, 2162-2169, (2012) · Zbl 1244.05186  Karoński, M; Łuczak, T; Thomason, A, Edge weights and vertex colours, J. Combin. Theory Ser. B, 91, 151-157, (2004) · Zbl 1042.05045  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 · Zbl 1325.05083  Li, H; Liu, B; Wang, G, Neighor sum distinguishing total colorings of $$K$$_{4}-minor free graphs, Front. Math. China, 8, 1351-1366, (2013) · Zbl 1306.05066  Pilśniak, M., Woźniak, M.: On the total-neighbor-distinguishing index by sums. Graph and Combin., DOI 10.1007/s00373-013-1399-4 · Zbl 1303.05058  Przybyło, J.: Irregularity strength of regular graphs. Electron. J. Combin., 15(1), #R82, 10pp (2008) · Zbl 1163.05329  Przybyło, J, Linear bound on the irregularity strength and the total vertex irregularity strength of graphs, SIAM J. Discrete Math., 23, 511-516, (2009) · Zbl 1216.05135  Przybyło, J; Woźniak, M, On a 1, 2 conjecture, Discrete Math. Theor. Comput. Sci., 12, 101-108, (2010) · Zbl 1250.05093  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 · Zbl 1302.05059  Wang, W., Huang, D.: The adjacent vertex distinguishing total coloring of planar graphs. J. Comb. Optim., DOI 10.1007/s10878-012-9527-2 · Zbl 1319.90076  Wang, W; Wang, P, On adjacent-vertex-distinguishing total coloring of $$K$$_{4}-minor free graphs, Sci. China, Ser. A, 39, 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) · Zbl 1209.05087  Wong, T; Zhu, X, Total weight choosability of graphs, J. Graph Theory, 66, 198-212, (2011) · Zbl 1228.05161  Wong, T; Zhu, X, Antimagic labelling of vertex weighted graphs, J. Graph Theory, 3, 348-359, (2012) · Zbl 1244.05192  Zhang, Z; Chen, X; Li, J; etal., On adjacent-vertex-distinguishing total coloring of graphs, Sci. China, Ser. A, 48, 289-299, (2005) · Zbl 1080.05036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.