Total weight choosability of Mycielski graphs

Abstract

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