Every nice graph is (1,5)-choosable

Abstract

A graph G=(V,E) is total weight (k,k′)-choosable if the following holds: For any list assignment L which assigns to each vertex v a set L(v) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a proper L-total weighting, i.e., a map ϕ:V∪E→R such that ϕ(z)∈L(z) for z∈V∪E, and ∑e∈E(u)ϕ(e)+ϕ(u)≠∑e∈E(v)ϕ(e)+ϕ(v) for every edge {u,v}. A graph is called nice if it contains no isolated edges. As a strengthening of the famous 1-2-3 conjecture, it was conjectured in Wong and Zhu (2011) [23] that every nice graph is total weight (1,3)-choosable. The problem whether there is a constant k such that every nice graph is total weight (1,k)-choosable remained open for a decade and was recently solved by Cao (2021) [6], who proved that every nice graph is total weight (1,17)-choosable. This paper improves this result and proves that every nice graph is total weight (1,5)-choosable.