Neighbor-sum-distinguishing edge choosability of subcubic graphs

Abstract

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 $\hbox {mad}(G)<\frac{5}{2}$.

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.