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Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13

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Abstract

A graph is IC-planar if it admits a drawing on the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring \(\phi \)\(V(G)~\cup ~E(G)\rightarrow \{1,2,\ldots ,k\}\) is called neighbor sum distinguishing if \(f_{\phi }(u)\ne f_{\phi }(v)\) for each \(uv\in E(G)\), where \(f_{\phi }(u)\) is the sum of the color of u and the colors of the edges incident with u. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by \(\chi ''_{\Sigma }(G)\). Pilśniak and Woźniak conjectured \(\chi _{\Sigma }''(G)\le \Delta (G)+3\) for any simple graph with maximum degree \(\Delta (G)\). This conjecture was confirmed for IC-planar graph with maximum degree at least 14. In this paper, by using the discharging method, we prove that this conjecture holds for any IC-planar graph G with \(\Delta (G)=13\).

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References

  • Albertson MO (2008) Chromatic number, independence ratio, and crossing number. Ars Math Contemp 1:1–6

    Article  MathSciNet  Google Scholar 

  • Alon N (1999) Combinatorial nullstellensatz. Comb Probab Comput 8:7–29

    Article  MathSciNet  Google Scholar 

  • Bondy JA, Murty USR (1976) Graph theory with applications. North-Holland, New York

    Book  Google Scholar 

  • Cheng X, Huang D, Wang G, Wu J (2015) Neighbor sum distinguishing total colorings of planar graphs with maximum degree \(\Delta \). Discrete Appl Math 190:34–41

    Article  MathSciNet  Google Scholar 

  • Ding L, Wang G, Yan G (2014) Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz. Sci China Math 57:1875–1882

    Article  MathSciNet  Google Scholar 

  • Ding L, Wang G, Wu J, Yu J (2017) Neighbor sum (set) distinguishing total choosability via the Combinatorial Nullstellensatz. Graphs Combin 33:885–900

    Article  MathSciNet  Google Scholar 

  • Dong A, Wang G (2014) Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree. Acta Math Sin (Engl Ser) 30:703–709

    Article  MathSciNet  Google Scholar 

  • Li H, Ding L, Liu B, Wang G (2015) Neighbor sum distinguishing total coloring of planar graphs. J Comb Optim 30(3):675–688

    Article  MathSciNet  Google Scholar 

  • Li H, Liu B, Wang G (2013) Neighbor sum distinguishing total colorings of \(K_{4}\)-minor free graphs. Front Math China 8(6):1351–1366

    Article  MathSciNet  Google Scholar 

  • Loeb S, Tang Y (2017) Asymptotically optimal neighbor sum distinguishing total colorings of graphs. Discrete Math 340(2):58–62

    Article  MathSciNet  Google Scholar 

  • Lu Y, Han M, Luo R (2018) Neighbor sum distinguishing total coloring and list neighbor sum distinguishing total coloring. Discrete Appl Math 237:109–115

    Article  MathSciNet  Google Scholar 

  • Pilśniak M, Woźniak M (2015) On the total-neighbor-distinguishing index by sums. Graphs Comb 31(3):771–782

    Article  MathSciNet  Google Scholar 

  • Qu C, Wang G, Wu J, Yu X (2016) Neighbor sum distinguishing total choosability of planar graphs. J Comb Optim 32:906–916

    Article  MathSciNet  Google Scholar 

  • Qu C, Wang G, Wu J, Yu X (2016) On the neighbor sum distinguishing total coloring of planar graphs. Theor Comput Sci 609:162–170

    Article  MathSciNet  Google Scholar 

  • Song W, Miao L, Duan Y (2018a) Neighbor sum distinguishing total choosability of IC-planar graphs. Discuss Math Graph Theory. https://doi.org/10.7151/dmgt.2145

    Article  Google Scholar 

  • Song W, Miao L, Li J, Zhao Y, Pang J (2018b) Neighbor sum distinguishing total coloring of sparse IC-planar graphs. Discrete Appl Math 239:183–192

    Article  MathSciNet  Google Scholar 

  • Song H, Pan W, Gong X, Xu C (2016) A note on the neighbor sum distinguishing total coloring of planar graphs. Theor Comput Sci 640:125–129

    Article  MathSciNet  Google Scholar 

  • Song H, Xu C (2017) Neighbor sum distinguishing total chromatic number of \(K_4\)-minor free graph. Front Math China 12(4):937–947

    Article  MathSciNet  Google Scholar 

  • Wang J, Cai J, Ma Q (2016) Neighbor sum distinguishing total choosability of planar graphs without 4-cycles. Discrete Appl Math 206:215–219

    Article  MathSciNet  Google Scholar 

  • Xu C, Li J, Ge S (2018) Neighbor sum distinguishing total chromatic number of planar graphs. Appl Math Comput 332:189–196

    MathSciNet  MATH  Google Scholar 

  • Yang D, Yu X, Sun L, Wu J, Zhou S (2017) Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10. Appl Math Comput 314:456–468

    MathSciNet  MATH  Google Scholar 

  • Yao J, Yu X, Wang G, Xu C (2016) Neighbor sum (set) distinguishing total choosability of \(d\)-degenerate graphs. Graphs Comb 32(4):1611–1620

    Article  MathSciNet  Google Scholar 

  • Zhang X, Hou J, Liu G (2015) On total colorings of 1-planar graphs. J Comb Optim 30(1):160–173

    Article  MathSciNet  Google Scholar 

  • Zhang X, Wu J, Liu G (2012) List edge and list total coloring of 1-planar graphs. Front Math China 7(5):1005–1018

    Article  MathSciNet  Google Scholar 

  • Zhang X, Wu J (2011) On edge colorings of 1-planar graphs. Inf Process Lett 111:124–128

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referees for their valuable comments. This work was supported by the National Natural Science Foundation of China (11671232).

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Correspondence to Changqing Xu.

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Song, C., Xu, C. Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13. J Comb Optim 39, 293–303 (2020). https://doi.org/10.1007/s10878-019-00467-1

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