Neighbor-sum-distinguishing edge choosability of subcubic graphs

A graph G is said to be neighbor-sum-distinguishing edge k-choose if, for every list L of colors such that L(e) is a set of k positive real numbers for every edge e, there exists a proper edge coloring which assigns to each edge a color from its list so that for each pair of adjacent vertices u and v 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. Let \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\) denote the smallest integer k such that G is neighbor-sum-distinguishing edge k-choose. In this paper, we prove that if G is a subcubic graph with the maximum average degree mad(G), then (1) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 7\); (2) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 6\) if \(\hbox {mad}(G)<\frac{36}{13}\); (3) \(\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 5\) if \(\hbox {mad}(G)<\frac{5}{2}\).

Authors and Affiliations

1. Department of Mathematics, Soochow University, Suzhou, 215006, China

   Jingjing Huo

2. Department of Mathematics, Hebei University of Engineering, Handan, 056038, China

   Jingjing Huo

3. School of Management, Beijing University of Chinese Medicine, Beijing, 100029, China

   Yiqiao Wang

4. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

   Weifan Wang

Corresponding author

Correspondence to Weifan Wang.

Jingjing Huo is research supported by NSFC (No. 11501161) and NSFHB (No. A2016402164). Yiqiao Wang is research supported by NSFC (No. 11301035) and (No. 11671053). Weifan Wang is research supported by NSFC (No. 11371328).

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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