Tag: SuilO

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.

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.