Total Weight Choosability of Cone Graphs

Abstract

A total weighting of a graph G is a mapping $\varphi$ that assigns a weight to each vertex and each edge of G. The vertex-sum of $v \in V(G)$ with respect to $\varphi$ is $S_{\varphi }(v)=\sum _{e\in E(v)}\varphi (e)+\varphi (v)$. A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph $G=(V,E)$ is called $(k,k{^{\prime }})$-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 $k{^{\prime }}$ real numbers, there is a proper total weighting $\varphi$ with $\varphi (y)\in L(y)$ for any $y \in V \cup E$. In this paper, we prove that for any graph $G\ne K_1$, 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 $\theta$-graph, then the m-cone graph of G is (1, 3)-choosable.