Minimum Degree Threshold for H-factors with High Discrepancy

Given a graph H, a perfect H-factor in a graph G is a collection of vertex-disjoint copies of H spanning G. Kuhn and Osthus showed that the minimum degree threshold for a graph G to contain a perfect H-factor is either given by 1-1/chi(H) or by 1-1/chi(cr)(H) depending on certain natural divisibility considerations. Given a graph G of order nn, a 2-edge-coloring of G and a subgraph G ' of G, we say that G ' has high discrepancy if it contains significantly (linear in n) more edges of one color than the other. Balogh, Csaba, Pluh & aacute;r and Treglown asked for the minimum degree threshold guaranteeing that every 2-edge-coloring of G has an H-factor with high discrepancy and they settled the case where H is a clique. Here we completely resolve this question by determining the minimum degree threshold for high discrepancy of H-factors for every graph H.