Friday, April 10 2020
12:30pm - 1:30pm
Department Seminar,Discrete Mathematics Seminar
Finding disjoint doubly chorded cycles

Abstract: In 1963, Corr\'{a}di and Hajnal verified a conjecture of Erd\H{o}s by showing that for all $k \in \mathbb{Z}^+$, if a graph $G$ has at least $3k$ vertices and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles. This result, which is best possible, has served as motivation behind many recent results in finding sharp minimum degree conditions that guarantee the existence of a variety of structures. In particular, Qiao and Zhang in 2010, showed that if $G$ has at least $4k$ vertices and $\delta(G) \ge \lfloor \frac{7k}{2}\rfloor$, then $G$ contains $k$ disjoint doubly chorded cycles. Then in 2015, Gould, Hirohata, and Horn proved that if $G$ has at least $6k$ vertices, then $\delta(G) \ge 3k$ is sufficient to guarantee $k$ disjoint doubly chorded cycles. In this talk, we extend the result of Gould et al. by showing that $\delta(G) \ge 3k$ suffices for graphs on at least $5k$ vertices (which is best possible for the given minimum degree), and we present an improvement of the Qiao and Zhang condition to $\delta(G) \ge \lceil \frac{10k-1}{3}\rceil$, which is sharp. This is joint work with Maia Wichman.
Speaker:Michael Santana
Affiliation:Grand Valley State University, Allendale, MI
Location:See Zoom link from email

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