An Improved Upper Bound on Edge Weight Choosability of Graphs.

The famous 1-2-3 conjecture due to Karoński, Łuczak and Thomason states that the edges of any nice graph (without a component isomorphic to $$K_{2}$$ ) may be weighted from the set {1,2,3} so that the vertices are properly coloured by the sums of their incident edge weights. Bartnicki, Grytczuk and Niwcyk introduced its list version. Assign to each edge $$e\in E(G)$$ a list of $$k$$ real numbers, say $$L(e)$$ , and choose a weight $$w(e)\in L(e)$$ for each $$e \in E(G)$$ . The resulting function $$w: E(G)\rightarrow \bigcup _{e\in E(G)}L(e)$$ is called an edge $$k$$ -list-weighting. Given a graph $$G$$ , the smallest $$k$$ such that any assignment of lists of size $$k$$ to $$E(G)$$ permits an edge $$k$$ -list-weighting which is a vertex coloring by sums is denoted by $$ch^{e}_{\Sigma }(G)$$ and called the edge weight choosability of $$G$$ . Bartnicki, Grytczuk and Niwcyk conjectured that if $$G$$ is a nice graph, then $$ch^{e}_{\Sigma }(G)\le 3$$ . There is no known constant $$K$$ such that $$ch^{e}_{\Sigma }(G) \le K$$ for any nice graph $$G$$ . Fu et al. proved that for a nice graph $$G$$ with maximum degree $$\Delta (G)$$ , $$ch^{e}_{\Sigma }(G)\le \lceil \frac{3\Delta (G)}{2}\rceil $$ . In this paper, we improve this bound to $$\lceil \frac{4\Delta (G)+8}{3}\rceil $$ . [ABSTRACT FROM AUTHOR]