An Improved Upper Bound on Edge Weight Choosability of Graphs

Abstract

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 $\lceil \frac{4\Delta (G)+8}{3}\rceil$.