After more experimentation, I finally cured my problem with...
SolidBrush WhiteBrush = new SolidBrush(Color.White);
myGraphics.FillRectangle(WhiteBrush,0,0,5000,5000);
i.e. I gave my graphics a solid white background before I drew any lines. Before I was drawing a black line on a NULL background. I have no idea why this would affect anything, but it did!