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:V∪E→ℝ such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑
y∈N(u)
f(uy)+f(u)≠ ∑
x∈N(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 s≥2) 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.
