Abstract
A proper edge coloring of a graph G using the color set \(\{1,2,\ldots , k\}\) is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by \(\chi '_{_{\sum }}(G)\). In this paper, we show that \(\chi '_{_{\sum }}(G)\le 6\) for any simple subcubic graph G. This improves a result in Flandrin et al. (Graphs Combin 29:1329–1336, 2013), which says that every cubic graph G has \(\chi '_{_{\sum }}(G)\le 8\).
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References
Akbari, S., Bidkhori, H., Nosrati, N.: \(r\)-Strong edge colorings of graphs. Discrete Math. 306, 3005–3010 (2006)
Alon, N.: Combinatorial Nullstellensatz. Combin. Probab. Comput. 8, 7–29 (1999)
Balister, P.N., Győri, E., Lehel, J., Schelp, R.H.: Adjacent vertex distinguishing edge-colorings. SIAM J. Discrete Math. 21, 237–250 (2007)
Bonamy, M., Przybyło, J.: On the neighbor sum distinguishing index of planar graphs. arxiv:1408.3190v1 (2014)
Flandrin, E., Marczyk, A., Przybyło, J., Saclé, J.F., Woźniak, M.: Neighbor sum distinguishing index. Graphs Combin. 29, 1329–1336 (2013)
Hatami, H.: \(\Delta + 300\) is a bound on the adjacent vertex distinguishing edge chromatic number. J. Combin. Theory Ser. B 95, 246–256 (2005)
Horňák, M., Huang, D., Wang, W.: On neighbor-distinguishing index of planar graphs. J. Graph Theory 76, 262–278 (2014)
Przybyło, J.: Asymptotically optimal neighbour sum distinguishing colourings of graphs. Random Struct. Algorithms 47, 776–791 (2015)
Przybyło, J.: Neighbor distinguishing edge colorings via the Combinatorial Nullstellensatz. SIAM J. Discrete Math. 27, 1313–1322 (2013)
Przybyło, J., Wong, T.: Neighbor distinguishing edge colorings via the Combinatorial Nullstellensatz Revisited. J. Graph Theory 80, 299–312 (2015)
Wang, G., Chen, Z., Wang, J.: Neighbor sum distinguishing index of planar graphs. Discrete Math. 334, 70–73 (2014)
Wang, W., Huang, D.: A characterization on the adjacent vertex distinguishing index of planar graphs with large maximum degree. SIAM J. Discrete Math. 29, 2412–2431 (2015)
Wang, G., Yan, G.: An improved upper bound for the neighbor sum distinguishing index of graphs. Discrete Appl. Math. 175, 126–128 (2014)
Wang, Y., Wang, W., Huo, J.: Some bounds on the neighbor-distinguishing index of graphs. Discrete Math. 338, 2006–2013 (2015)
Zhang, Z., Liu, L., Wang, J.: Adjacent strong edge coloring of graphs. Appl. Math. Lett. 15, 623–626 (2002)
Zhang, L., Wang, W., Lih, K.-W.: An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph. Discrete Appl. Math. 162, 348–354 (2014)
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J. Huo’s research was supported by NSFC (No. 11501161)and NSFHB (No. A2016402164). W. Wang’s research was supported by NSFC (No. 11371328, No. 11471293) and ZJNSFC (No. LY14A010014).
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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Huo, J., Wang, W. & Xu, C. Neighbor Sum Distinguishing Index of Subcubic Graphs. Graphs and Combinatorics 33, 419–431 (2017). https://doi.org/10.1007/s00373-017-1760-0
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DOI: https://doi.org/10.1007/s00373-017-1760-0