LAUSR

A total weighting of a graph G is a mapping $$\phi $$ϕ that assigns a weight to each vertex and each edge of G. The vertex-sum of $$v \in V(G)$$v∈V(G) with respect to $$\phi $$ϕ is $$S_{\phi }(v)=\sum _{e\in E(v)}\phi (e)+\phi (v)$$Sϕ(v)=∑e∈E(v)ϕ(e)+ϕ(v). A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph $$G=(V,E)$$G=(V,E) is called $$(k,k')$$(k,k′)-choosable if the following is true: If each vertex x is assigned a set L(x) of k real numbers, and each edge e is assigned a set L(e) of $$k'$$k′ real numbers, then there is a proper total weighting $$\phi $$ϕ with $$\phi (y)\in L(y)$$ϕ(y)∈L(y) for any $$y \in V \cup E$$y∈V∪E. In this paper, we prove that for any graph $$G\ne K_1$$G≠K1, the Mycielski graph of G is (1,4)-choosable. Moreover, we give some sufficient conditions for the Mycielski graph of G to be (1,3)-choosable. In particular, our result implies that if G is a complete bipartite graph, a complete graph, a tree, a subcubic graph, a fan, a wheel, a Halin graph, or a grid, then the Mycielski graph of G is (1,3)-choosable.