For positive integers, $r \ge 3, h \ge 1,$ and $k \ge 1$, Bollobás, Saito, and Wormald proved some sufficient conditions for an $h$-edge-connected $r$-regular graph to have a k-factor in 1985. Lu gave an upper bound for the third-largest eigenvalue in a connected $r$-regular graph to have a $k$-factor in 2010. Gu found an upper bound for certain eigenvalues in an $h$-edge-connected $r$-regular graph to have a $k$-factor in 2014. For positive integers $a \le b$, an even (or odd) $[a, b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for each vertex $v \in V (G)$, $d_H(v)$ is even (or odd) and $a \le d_H(v) \le b$. In this talk, we provide best upper bounds (in terms of $a, b$, and $r$) for certain eigenvalues (in terms of $a, b, r$, and $h$) in an $h$-edge-connected $r$-regular graph $G$ to guarantee the existence of an even $[a, b]$-factor or an odd $[a, b]$-factor. This result extends the one of Bollobás, Saito, and Wormald, the one of Lu, and the one of Gu.

## Suil O (오수일) presented his work on an upper bound of the 3rd largest eigenvalue of a regular graph to guarantee the existence of an odd (1,b)-factor at the discrete math seminar

On June 19, 2019, Suil O (오수일) from SUNY Korea, Incheon presented his work on an upper bound of the third largest eigenvalue of a connected r-regular graph to have an odd (1,b)-factor. The title of his talk was “An odd [1,b]-factor in regular graphs from eigenvalues”.

## Suil O (오수일), An odd [1,b]-factor in regular graphs from eigenvalues

An odd $[1,b]$-factor of a graph is a spanning subgraph $H$ such that for every vertex $v \in V(G)$, $1 \le d_H(v) \le b$, and $d_H(v)$ is odd. For positive integers $r \ge 3$ and $b \le r$, Lu, Wu, and Yang gave an upper bound for the third largest eigenvalue in an $r$-regular graph with even number of vertices to guarantee the existence of an odd [1,b]-factor.

In this talk, we improve their bound.