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Total weight choosability of graphs: Towards the 1-2-3-conjecture
Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2021-02-03 , DOI: 10.1016/j.jctb.2021.01.008
Lu Cao

Let G=(V,E) be a graph. A proper total weighting of G is a mapping w:VER such that the following sum for each vV:w(v)+eE(v)w(e) gives a proper vertex colouring of G. For any a,bN+, we say that G is total weight (a,b)-choosable if for any {Sv:vV}[R]a and {Se:vE}[R]b, there exists a proper total weighting w of G such that w(v)Sv for vV and w(e)Se for eE. 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.



中文翻译:

图的总重量选择率:朝向1-2-3-猜想

G=VË成为图。G的适当总权重是一个映射wVË[R 这样每个的以下总和 vVwv+ËËvwË给出G的适当顶点着色。对于任何一个bñ+,我们说G是总重量一个b-如果有的话可以选择 {小号vvV}[[R]一个{小号ËvË}[[R]b中,存在一个适当的总加权瓦特ģ使得wv小号v 对于 vVwË小号Ë 对于 ËË。1-2-3猜想的增强表明,每个没有孤立边的图都是总权重1个3-可选择。在本文中,我们证明了每个没有孤立边的图都是总权重1个17-可选择。我们还证明了二部图的总权重选择性的一些新结果。

更新日期:2021-02-03
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