Total weight choosability of graphs: Towards the 1-2-3-conjecture

Let G = (V, E) be a graph. A proper total weighting of G is a mapping w : V boolean OR E -> R such that the following sum for each v is an element of V: w(v) + Sigma(e is an element of E(v)) w(e) gives a proper vertex colouring of G. For any a, b is an element of N+, we say that Gis total weight (a, b)-choosable if for any {S-v : v is an element of V} subset of [R](a) and {S-e: v is an element of E} subset of [R](b), there exists a proper total weighting w of G such that w(v) is an element of S-v for v is an element of V and w(e) is an element of S-e for e is an element of E. A strengthening of the 1-2-3 Conjecture states that every graph without an isolated edge is total weight (1, 3)-choosable. In this paper, we prove that every graph without an isolated edge is total weight (1, 17)-choosable. We also prove some new results on the total weight choosability of bipartite graphs. (C) 2021 Elsevier Inc. All rights reserved.