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

• Published in: The Electronic journal of combinatorics Vol. 31; no. 3
• Main Authors: Bradač, Domagoj, Christoph, Micha, Gishboliner, Lior
• Format: Journal Article
• Language: English
• Published: 20.09.2024
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/12145

Given a graph $H$, a perfect $H$-factor in a graph $G$ is a collection of vertex-disjoint copies of $H$ spanning $G$. Kühn 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 $n$, 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, Pluhár 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 $H$.