Every graph is \((2,3)\)-choosable (Q519997)

A proper total weighting of a graph \(G=(V,E)\) is a function \(f:V\cup E\rightarrow N\) such that \(\sum\limits_{e=E(u)}f(u)+f(e)\neq \sum\limits_{e\in E(v)}f(v)+f(e)\) for every edge \(e=uv\) of \(G.\) The problem is to determine the smallest integer \(k\) such that there exists a proper total weighting \(f\) with \(f(z)\in \{1,\dots,k\}.\) This problem is a generalization of the famous \(1-2-3\) conjecture. In this paper, the authors study a ``choosabilty'' version of the proper total weighting problem. It is proved there that if each edge \(e\) is assigned a set \(L(e)\) with 3 weights and each vertex \(v\) with a set \(L(v)\) of two weights then there is a proper total weighing of \(G\) where the weight of each element \(z\) in \(V\cup E\) is taken from the set \(L(z).\)