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Total coloring of outer-1-planar graphs with near-independent crossings

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Abstract

A graph G is outer-1-planar with near-independent crossings if it can be drawn in the plane so that all vertices are on the outer face and \(|M_G(c_1)\cap M_G(c_2)|\le 1\) for any two distinct crossings \(c_1\) and \(c_2\) in G, where \(M_G(c)\) consists of the end-vertices of the two crossed edges that generate c. In Zhang and Liu (Total coloring of pseudo-outerplanar graphs, arXiv:1108.5009), it is showed that the total chromatic number of every outer-1-planar graph with near-independent crossings and with maximum degree at least 5 is \(\Delta +1\). In this paper we extend the result to maximum degree 4 by proving that the total chromatic number of every outer-1-planar graph with near-independent crossings and with maximum degree 4 is exactly 5.

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References

  • Auer C, Bachmaier C, Brandenburg FJ et al (2016) Outer 1-planar graphs. Algorithmica 74(4):1293–1320

    Article  MathSciNet  MATH  Google Scholar 

  • Behzad M (1965) Graphs and their chromatic numbers. Doctoral thesis, Michigan State University

  • Eggleton RB (1986) Rectilinear drawings of graphs. Utilitas Math 29:149–172

    MathSciNet  MATH  Google Scholar 

  • Vizing V (1968) Some unsolved problems in graph theory. Uspekhi Mat Nauk 23:117–134

    MathSciNet  MATH  Google Scholar 

  • Wang W, Zhang K (1999) \(\Delta \)-Matchings and edge-face chromatic numbers. Acta Math Appl Sin 22:236–242

    MathSciNet  MATH  Google Scholar 

  • Wu JL, Hu D (2004) Total coloring of series-parallel graphs. ARS Comb 73:209–211

    MathSciNet  MATH  Google Scholar 

  • Yap HP (1996) Total colourings of graphs. Lecture notes in mathematics 1623. Springer, Berlin

  • Zhang X (2013) List total coloring of pseodo-outerplanar graphs. Discret Math 313:2297–2306

    Article  MATH  Google Scholar 

  • Zhang X (2014) Drawing complete multipartite graphs on the plane with restrictions on crossings. Acta Math Sin (Engl Ser) 30(12):2045–2053

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang X, Liu G Total coloring of pseudo-outerplanar graphs. arXiv:1108.5009

  • Zhang X, Liu G, Wu JL (2012) Edge covering pseudo-outerplanar graphs with forests. Discret Math 312:2788–2799

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Z, Zhang J, Wang J (1988) The total chromatic number of some graphs. Sci Sin Ser A 31:1434–1441

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author appreciates the anonymous referees sincerely for their helpful comments. This paper is supported by National Natural Science Foundation of China (No. 11301410), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130203120021), and the Fundamental Research Funds for the Central Universities (No. JB150714).

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Correspondence to Xin Zhang.

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Zhang, X. Total coloring of outer-1-planar graphs with near-independent crossings. J Comb Optim 34, 661–675 (2017). https://doi.org/10.1007/s10878-016-0093-x

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