Abstract
The neighbor-distinguishing total coloring of a graph G is a proper total coloring of G using k colors such that any two adjacent vertices have different sets of colors. It was known that every planar graph G with \(\Delta \ge 10\) is neighbor-distinguishing totally \((\Delta +3)\)-colorable. In this paper, we extend this result to the case \(\Delta =9\). Namely, we prove that every planar graph G with \(\Delta =9\) is neighbor-distinguishing totally 12-colorable.
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References
Alon N (1999) Combinatorial nullstellensatz. Comb Probab Comput 8:7–29
Appel K, Haken W (1976) Every planar map is four colourable. Bull Am Math Soc 82:711–712
Chen X (2008) On the adjacent vertex distinguishing total coloring numbers of graphs with \(\Delta =3\). Discrete Math 308:4003–4007
Cheng X, Wang G, Wu J (2017) The adjacent vertex distinguishing total chromatic numbers of planar graphs with \(\Delta =10\). J Comb Optim 34:383–397
Coker T, Johannson K (2012) The adjacent vertex distinguishing total chromatic number. Discrete Math 312:2741–2750
Grötzsch H (1959) Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. Wiss Z Martin Luther-Univ Halle Wittenberg, Math-Nat Reihe 8:109–120
Huang D, Wang W, Yan C (2012) A note on the adjacent vertex distinguishing total chromatic number of graphs. Discrete Math 312:3544–3546
Huang D, Wang W (2012) Adjacent vertex distinguishing total colorings of planar graphs with large maximum degree. Sci Sin Math 42:151–164
Hulgan J (2009) Concise proofs for adjacent vertex-distinguishing total colorings. Discrete Math 309:2548–2550
Lu Y, Li J, Luo R, Miao Z (2017) Adjacent vertex distinguishing total coloring of graphs with maximum degree 4. Discrete Math 340:119–123
Sanders D, Zhao Y (2001) Planar graphs of maximum degree seven are Class I. J Comb Theory Ser B 83:201–212
Vizing VG (1964) On an estimate of the chromatic index of a p-graph. Diskret Anal 3:25–30
Wang H (2007) On the adjacent vertex distinguishing total chromatic number of the graphs with \(\Delta (G)=3\). J Comb Optim 14:87–109
Wang W, Wang Y (2008) Adjacent vertex distinguishing total coloring of graphs with lower average degree. Taiwanese J Math 12:979–990
Wang W, Wang P (2009) On adjacent-vertex-distinguishing total coloring of \(K_{4}\)-minor free graphs. Sci China Ser A 39:1462–1472
Wang W, Huang D (2014) The adjacent vertex distinguishing total coloring of planar graphs. J Comb Optim 27:379–396
Wang Y, Wang W (2010) Adjacent vertex distinguishing total colorings of outerplanar graphs. J Comb Optim 19:123–133
Zhang Z, Chen X, Li J, Yao B, Lu X, Wang J (2005) On adjacent-vertex-distinguishing total coloring of graphs. Sci China Ser A 48:289–299
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Weifan Wang: Research supported by NSFC (No. 11371328; 11771402). Jingjing Huo: Research supported by NSFC (No. 11701136; 11501161) and NSFHB (No. A2016402164). Danjun Huang: Research supported by NSFC (No. 11301486). Yiqiao Wang: Research supported by NSFC (No. 11671053).
Appendix
Appendix
In MATLAB, the program calculating \(\frac{\partial ^{k_{1}+k_{2}+\cdots +k_{n}}Q}{\partial x_{1}^{k_{1}}\partial x_{2}^{k_{2}}\cdots \partial x_{n}^{k_{n}}}\) is given as follows.
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Wang, W., Huo, J., Huang, D. et al. Planar graphs with \(\Delta =9\) are neighbor-distinguishing totally 12-colorable. J Comb Optim 37, 1071–1089 (2019). https://doi.org/10.1007/s10878-018-0334-2
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DOI: https://doi.org/10.1007/s10878-018-0334-2
Keywords
- Planar graph
- Neighbor-distinguishing total coloring
- Maximum degree
- Combinatorial Nullstellensatz
- Discharging