On total weight choosability of graphs

For a graph G with vertex set V and edge set E, a (k,k')-total list assignment L of G assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k' real numbers as permissible weights. If for any (k,k')-total list assignment L of G, there exists a mapping f:Va(a)E -> a"e such that f(y)aL(y) for each yaVa(a)E, and for any two adjacent vertices u and v, a (yaN(u)) f(uy)+f(u)not equal a (xaN(v)) f(vx)+f(v), then G is (k,k')-total weight choosable. It is conjectured by Wong and Zhu that every graph is (2,2)-total weight choosable, and every graph with no isolated edges is (1,3)-total weight choosable. In this paper, it is proven that a graph G obtained from any loopless graph H by subdividing each edge with at least one vertex is (1,3)-total weight choosable and (2,2)-total weight choosable. It is shown that s-degenerate graphs (with sa parts per thousand yen2) are (1,2s)-total weight choosable. Hence planar graphs are (1,10)-total weight choosable, and outerplanar graphs are (1,4)-total weight choosable. We also give a combinatorial proof that wheels are (2,2)-total weight choosable, as well as (1,3)-total weight choosable.