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Tuesday, May 19, 2020

**Abstract:** Call a blowup of a graph an $n$-blowup if each part has size $n$. For a subgraph $G$ of a blowup of $F$, we define the minimum partial degree of $G$ to be the smallest minimum degree over all of the bipartite subgraphs of $G$ that correspond to edges of $F$. Johannson proved that when $G$ is a spanning subgraph of the $n$-blowup of a triangle with minimum partial degree at least $2n/3 + n^{1/2}$, then G contains $n$ vertex disjoint triangles. Fischer's Conjecture, which was proved by Keevash and Mycroft in 2015, is a generalization of this result to complete graphs larger than the triangle. Another generalization, conjectured independently by Fischer and Häggkvist, is the following: If $G$ is a spanning subgraph of the $n$-blowup of $C_k$ with minimum partial degree at least $(1 + 1/k)n/2 + 1$, then $G$ contains $n$ vertex disjoint copies of $C_k$ that each intersect all of the $k$ parts. In this talk, we will discuss a proof of an asymptotic version of this conjecture.

This is joint work with Beka Ergemlidze.

Please email Sean English at SEnglish (at) Illinois (dot) edu for the Zoom ID and password.