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Given a graph $H$, a perfect $H$-factor in a graph $G$ is a collection ofvertex-disjoint copies of $H$ spanning $G$. K\"uhn and Osthus showed that theminimum degree threshold for a graph $G$ to contain a perfect $H$-factor iseither given by $1-1/\chi(H)$ or by $1-1/\chi_{cr}(H)$ depending on certainnatural divisibility considerations. Given a graph $G$ of order $n$, a$2$-edge-coloring of $G$ and a subgraph $G'$ of $G$, we say that $G'$ has highdiscrepancy if it contains significantly (linear in $n$) more edges of onecolor than the other. Balogh, Csaba, Pluh\'ar and Treglown asked for theminimum 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 aclique. Here we completely resolve this question by determining the minimumdegree threshold for high discrepancy of $H$-factors for every graph $H$.

Minimum Degree Threshold for $H$-factors with High Discrepancy