Total weight choosability of Mycielski graphs

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 is an element of V(G) with respect to phi is S-phi(v) = Sigma(e is an element of E(v))phi(e) + phi(v) . A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph is 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) is an element of L(y) for any y is an element of V boolean OR E . In this paper, we prove that for any graph G not equal K-1,K- 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.