Transversal $C_k$-factors in subgraphs of the balanced blowup of $C_k$

Graph Theory Seminar
Tuesday, November 17, 2020 - 3:45pm for 1 hour (actually 50 minutes)
Location For password, please email Anton Bernshteyn (bahtoh ~at~
Theo Molla – University of South Florida – molla@usf.edu
Anton Bernshteyn

Call a blowup of a graph $F$ 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 the bipartite subgraphs of $G$ that correspond to edges of $F$. Johannson proved that if the minimum partial degree of a spanning subgraph of the $n$-blowup of a triangle is $2n/3 + n^{1/2}$, then it contains a collection of $n$ 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 $(1 + 1/k)n/2 + 1$, then $G$ contains $n$ vertex disjoint copies of $C_k$ that each intersect each of the $k$ parts. In this talk, we will show that this conjecture holds asymptotically. We will also discuss related conjectures and results. 

This is joint work with Beka Ergemlidze.