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', ₄ ₄ (ₕ) f (e) +f (v) ₄ ₄ (ₕ') 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.