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Neighbor Sum Distinguishing Index of \(K_4\)-Minor Free Graphs

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Abstract

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors from \([k]=\{1,2,\ldots ,k\}\). A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge \(uv\in E(G)\), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By nsdi(G), we denote the smallest value k in such a coloring of G. It was conjectured by Flandrin et al. that if G is a connected graph with at least three vertices and \(G\ne C_5\), then nsdi\((G)\le \varDelta (G)+2\). In this paper, we prove that this conjecture holds for \(K_4\)-minor free graphs, moreover if \(\varDelta (G)\ge 5\), we show that nsdi\((G)\le \varDelta (G)+1\). The bound \(\varDelta (G)+1\) is sharp.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (71201093, 11471193,11401266,11371355), the Independent Innovation Foundation of Shandong University (IFYT14011, IFYT14012), the Foundation for Distinguished Young Scholars of Shandong Province (JQ201501) and the Fundamental Research Funds of Shandong University.

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Authors and Affiliations

  1. School of Management, Shandong University, 250100, Jinan, Shandong, People’s Republic of China

    Jianghua Zhang

  2. School of Mathematics, Shandong University, 250100, Jinan, Shandong, People’s Republic of China

    Laihao Ding & Guanghui Wang

  3. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 10080, People’s Republic of China

    Guiying Yan

  4. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, People’s Republic of China

    Shan Zhou

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  1. Jianghua Zhang
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  2. Laihao Ding
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  3. Guanghui Wang
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  4. Guiying Yan
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  5. Shan Zhou
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Correspondence to Guanghui Wang.

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Zhang, J., Ding, L., Wang, G. et al. Neighbor Sum Distinguishing Index of \(K_4\)-Minor Free Graphs. Graphs and Combinatorics 32, 1621–1633 (2016). https://doi.org/10.1007/s00373-015-1655-x

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  • DOI: https://doi.org/10.1007/s00373-015-1655-x

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