We study online robust matrix completion on graphs. At each iteration a vector with some entries missing is revealed and our goal is to reconstruct it by identifying the underlying low-dimensional subspace from which the vectors are drawn. We assume there is an underlying graph structure to the data, that is, the components of each vector correspond to nodes of a certain known and fixed graph, and their values are related accordingly. We propose an algorithm that exploits the graph to reconstruct the incomplete data, in the scenario where there is outlier noise. The theoretical properties of the algorithms are studied and numerical experiments using both synthetic and real world datasets verify the improved performance of the proposed technique compared to other state of the art algorithms.