Abstract
We consider proper edge colorings of a graph G using colors in \(\{1,\ldots ,k\}\). Such a coloring is called neighbor sum distinguishing if for each pair of adjacent vertices u and v, the sum of the colors of the edges incident with u is different from the sum of the colors of the edges incident with v. The smallest value of k in such a coloring of G is denoted by \({\mathrm ndi}_{\Sigma }(G)\). In this paper we show that if G is a 2-degenerate graph without isolated edges, then \({\mathrm ndi}_{\Sigma }(G)\le \max \{\Delta (G)+2,7\}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akbari S, Bidkhori H, Nosrati N (2006) \(r\)-Strong edge colorings of graphs. Discrete Math. 306:3005–3010
Balister PN, Győri E, Lehel J, Schelp RH (2007) Adjacent vertex distinguishing edge-colorings. SIAM J Discrete Math 21:237–250
Basavaraju M, Chandran LS (2012) Acyclic edge coloring of 2-degenerate graphs. J Graph Theory 69:1–27
Bonamy M, Przybyło J (2016) On the neighbour sum distinguishing index of planar graphs. J Graph Theory. doi:10.1002/jgt.22098
Chang GJ, Narayanan N (2013) Strong chromatic index of 2-degenerate graphs. J Graph Theory 73:119–126
Dong AJ, Wang GH (2012) Neighbor sum distinguishing coloring of some graphs. Discrete Math Algorithms Appl 4(4):125–147
Dong AJ, Wang GH, Zhang JH (2014) Neighbor sum distinguishing edge colorings of graphs with bounded maximum average degree. Discrete Appl Math 166:84–90
Edwards K, Horňák M, Woźniak M (2006) On the neighbour-distinguishing index of a graph. Graphs Comb 22:341–350
Flandrin E, Marczyk A, Przybyło J, Saclé J-F, Woźniak M (2013) Neighbor sum distinguishing index. Graphs Comb 29:1329–1336
Hatami H (2005) \(\Delta +300\) is a bound on the adjacent vertex distinguishing edge chromatic number. J Comb Theory Ser B 95:246–256
Horňák M, Huang DJ, Wang WF (2014) On neighbor-distinguishing index of planar graphs. J Graph Theory 76(4):262–278
Hu XL, Chen YJ, Luo R, Miao ZK (2015) Neighbor sum distinguishing edge colorings of sparse graphs. Discrete Appl Math 193:119–125
Kalkowski M, Karoński M, Pfender F (2010) Vertex coloring edge-weightings: towards the 1–2–3-conjecture. J Comb Theory Ser B 100:347–349
Karoński M, Łuczak T, Thomasson A (2004) Edge weights and vertex colors. J Comb Theory Ser B 91:151–157
Przybyło J (2013) Neighbor distinguishing edge colorings via the combinatorial nullstellensatz. SIAM J Discrete Math 27(3):1313–1322
Przybyło J (2015) Asymptotically optimal neighbour sum distinguishing colourings of graphs. Random Struct Algorithms 47(4):776–791
Seamone B (2012) The 1–2–3 conjecture and related problems: a survey. arXiv:1211.5122v1
Wang GH, Chen ZM, Wang JH (2014) Neighbor sum distinguishing index of planar graphs. Discrete Math 334:70–73
Wang Y, Cheng J, Luo R, Mulley G (2016) Adjacent vertex-distinguishing edge coloring of 2-degenerate graphs. J Comb Optim 143(2):1–7
Zhang ZF, Liu LZ, Wang JF (2002) Adjacent strong edge coloring of graphs. Appl Math Lett 15:623–626
Acknowledgements
Many thanks to the anonymous referees for their careful comments that improved the presentation of this paper. The first author was supported by NSFC under Grant Number 11601176, the second author was supported by NSFC under Grant Numbers 11371193 and 11671198, and the fourth author was supported by NSFC under Grant Numbers 11171288 and 11571149.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, X., Chen, Y., Luo, R. et al. Neighbor sum distinguishing index of 2-degenerate graphs. J Comb Optim 34, 798–809 (2017). https://doi.org/10.1007/s10878-017-0110-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-017-0110-8