I'm using Photoshop to save illustrations with a transparent middle as a compressed png. file (18% of the original size). When compressing, thin grey lines appear around the transparent edge. These disappear when I select 'Nearest Neighbor' as the compression option, but it causes the image edges to become very jagged.
This isn't an output issue – you just need to finesse the transparent area of your image. Make sure there are no grey pixels stuck to the inside edge of your transparent area (you may want to use 'Select Colour Range' or the 'Refine Edge' tool to select those grey pixels) before you go to export.