## On Degree-Sums and Chorded Cycles

- Oct. 25, 2019
- 2:30 p.m.
- LeConte 312

## Abstract

The minimum degree of a graph is useful for determining the existence of certain subgraphs. For example, if $\delta(G) \geq 2$, then $G$ contains a cycle; if $\delta(G) \geq 3$, then $G$ contains a cycle with a chord. Considerable work has been done to improve these simple results, by finding degree conditions and degree-sum conditions that imply $k$ vertex-disjoint cycles or chorded cycles, for $k\geq 2$. We say that $\sigma _ t(G)$ is the minimum degree-sum of all independent $t$-sets in $G$. In this talk, I present joint work with Ron Gould, Kazuhide Hirohata, and Ariel Keller settling the following conjecture: For $k\geq 2$, $t\geq 5$, and $|G|$ sufficiently large, $\sigma _ t(G) \geq t(3k-1)+1$ implies that $G$ contains $k$ vertex-disjoint chorded cycles. To show this we investigate and characterize graphs without chorded cycles.