Total weight choosability of Mycielski graphs

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 $$S_{\phi }(v)=\sum _{e\in E(v)}\phi (e)+\phi (v)$$ S ϕ ( 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 ≠ K 1 , 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.