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Adjacent vertex distinguishing edge coloring of IC-planar graphs


The adjacent vertex distinguishing edge coloring of a graph G is a proper edge coloring in which each pair of adjacent vertices is assigned different color sets. The smallest number of colors for which G has such a coloring is denoted by \(\chi '_a(G)\). An important conjecture due to Zhang et al. (Appl Math Lett 15:623–626, 2002) asserts that \(\chi '_a(G)\le \Delta (G)+2\) for any connected graph G with order at least 6. By applying the discharging method, we show that this conjecture is true for any IC-planar graph G with \(\Delta (G)\ge 16\).

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

    MathSciNet  Article  Google Scholar 

  2. Bonamy M, Bousquet N, Hocquard H (2013) Adjacent vertex-distinguishing edge colorings of graphs. EuroComb 16:313–318

    MathSciNet  MATH  Google Scholar 

  3. Balister PN, Györi E, Lehel J, Schelp RH (2007) Adjacent vertex distinguishing edge-colorings. SIAM J Discrete Math 21(1):237–250

    MathSciNet  Article  Google Scholar 

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

    Book  Google Scholar 

  5. Horňák M, Huang DJ, Wang WF (2014) On neighbor-distinguishing index of planar graphs. J Graph Theory 76(4):262–278

    MathSciNet  Article  Google Scholar 

  6. Wang WF, Huang DJ (2015) A characterization on the adjacent vertex distinguishing index of planar graphs with large maximum degree. SIAM J Discrete Math 29(4):2412–2431

    MathSciNet  Article  Google Scholar 

  7. Yan CC, Huang DJ, Wang WF (2012) Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least four. J Math Study 45(4):331–341

    MathSciNet  MATH  Google Scholar 

  8. Zhang ZF, Liu LZ, Wang JF (2002) Adjacent strong edge coloring of graphs. Appl Math Lett 15:623–626

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

Download references


This work was supported by the National Natural Science Foundation of China (12071260, 12001154), and the Natural Science Foundation of Hebei Province(A2021202025).

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Correspondence to Changqing Xu.

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Liu, Z., Xu, C. Adjacent vertex distinguishing edge coloring of IC-planar graphs. J Comb Optim (2021).

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  • IC-planar graph
  • Adjacent vertex distinguishing edge coloring
  • Discharging method