The effect of edge structures in graphene sheets has been well investigated theoretically but most experimental demonstrations of the functionalization have been for the bulk structures because of only a few reports on chemical methods to modify the edges selectively. We herein report a chemical method using the Lemieux-von Rudloff reagent that selectively oxidizes only the edges of graphene sheets. The selective oxidation at the edges of the graphene sheet was confirmed by thermogravimetric analysis (TGA), Raman mapping measurements, and X-ray photoelectron spectroscopy (XPS). The TGA result of the oxidized graphite with different particle sizes showed a slight weight loss at approximately 350 degrees C (2.29% for the middle particles (35 microm)), which indicates thermal decomposition of the oxidized edge part. The Raman mapping measurement in the inner part of graphene sheets didn't detect any defects or translational symmetry breaking after the oxidation. The XPS data clearly showed that the total carbon atom content present as C--O, C==O, and O--C==C increased from 4.65 to 6.18% by the oxidation. Using the obtained edge-oxidized graphene as a starting material, various functionalizations of the edge structure are expected in the future.