An Improved Upper Bound on Edge Weight Choosability of Graphs

Abstract

The famous 1-2-3 conjecture due to Karonski, ?uczak and Thomason states that the edges of any nice graph (without a component isomorphic to $$K_{2}$$K2) 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 $$e\in E(G)$$e?E(G) a list of $$k$$k real numbers, say $$L(e)$$L(e), and choose a weight $$w(e)\in L(e)$$w(e)?L(e) for each $$e \in E(G)$$e?E(G). The resulting function $$w: E(G)\rightarrow \bigcup _{e\in E(G)}L(e)$$w:E(G)??e?E(G)L(e) is called an edge $$k$$k-list-weighting. Given a graph $$G$$G, the smallest $$k$$k such that any assignment of lists of size $$k$$k to $$E(G)$$E(G) permits an edge $$k$$k-list-weighting which is a vertex coloring by sums is denoted by $$ch^{e}_{\Sigma }(G)$$chΣe(G) and called the edge weight choosability of $$G$$G. Bartnicki, Grytczuk and Niwcyk conjectured that if $$G$$G is a nice graph, then $$ch^{e}_{\Sigma }(G)\le 3$$chΣe(G)≤3. There is no known constant $$K$$K such that $$ch^{e}_{\Sigma }(G) \le K$$chΣe(G)≤K for any nice graph $$G$$G. Fu et al. proved that for a nice graph $$G$$G with maximum degree $$\Delta (G)$$Δ(G), $$ch^{e}_{\Sigma }(G)\le \lceil \frac{3\Delta (G)}{2}\rceil $$chΣe(G)≤?3Δ(G)2?. In this paper, we improve this bound to $$\lceil \frac{4\Delta (G)+8}{3}\rceil $$?4Δ(G)+83?.