Total Weight Choosability of Cone Graphs

A total weighting of a graph G is a mapping $$\varphi $$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 $$\varphi $$φ is $$S_{\varphi }(v)=\sum _{e\in E(v)}\varphi (e)+\varphi (v)$$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 $$(k,k{^{\prime }})$$(k,k′)-choosable if the following is true: For any total 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 $$k{^{\prime }}$$k′ real numbers, there is a proper total weighting $$\varphi $$φ with $$\varphi (y)\in L(y)$$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, for any positive integer m, the m-cone graph of G is (1, 4)-choosable. Moreover, we give some sufficient conditions for the m-cone graph of G to be (1, 3)-choosable. In particular, if G is a tree, a complete bipartite graph or a generalized $$\theta $$θ-graph, then the m-cone graph of G is (1, 3)-choosable.