Weight Choosability of Graphs with Maximum Degree 4

Letkbe a positive integer. A graphGisk-weight choosable if, for any assignmentL(e) ofkreal numbers to eache is an element of E(G), there is a mappingf: E(G) -> Double-struck capital R such thatf(uv) is an element of L(uv) and n-ary sumation e is an element of partial differential (u)f(e)not equal n-ary sumation e is an element of partial differential (v)f(e)for eachuu is an element of E(G), where partial differential (v)is the set of edges incident withv. As a strengthening of the famous 1-2-3-conjecture, Bartnicki, Grytczuk and Niwcyk [Weight choosability of graphs.J. Graph Theory,60, 242-256 (2009)] conjecture that every graph without isolated edge is 3-weight choosable. This conjecture is wildly open and it is even unknown whether there is a constantksuch that every graph without isolated edge isk-weight choosable. In this paper, we show that every connected graph of maximum degree 4 is 4-weight choosable.