  We consider bond percolation on random graphs with given degrees and bounded
average degree. In particular, we consider the order of the largest component
after the random deletion of the edges of such a random graph. We give a rough
characterisation of those degree distributions for which bond percolation with
high probability leaves a component of linear order, known usually as a giant
component. We show that essentially the critical condition has to do with the
tail of the degree distribution. Our proof makes use of recent technique
introduced by Joos et al. [FOCS 2016, pp. 695--703], which is based on the
switching method and avoids the use of the classic configuration model as well
as the hypothesis of having a limiting object. Thus our results hold for sparse
degree sequences without the usual restrictions that accompany the
configuration model.
