Some results on total weight choosability

A graph $G=(V,E)$ is called $(k,k')$-choosable if 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 $k'$ real numbers, there is a mapping f:V∪ E→ R such that f(y)∈ L(y) for any y∈ V∪ E and for any two adjacent vertices $v, v'$, ∑e∈ E(v)f(e)+f(v)≠ ∑e∈ E(v')f(e)+f(v'), where $E(x)$ denotes the set of incident edges of a vertex x∈ V(G). In this paper, we characterize a sufficient condition on $(1,2)$-choosable of graphs. We show that every connected $(n,m)$-graph is both $(2,2)$-choosable and $(1,3)$-choosable if $m=n$ or $n+1$, where $(n,m)$-graph denotes the graph with n vertices and m edges. Furthermore, we prove that some graphs obtained by some graph operations are $(2,2)$-choosable.