\(k\)-Factor-Covered Regular Graphs

Abstract

A graph \(G\) is called \(f\)-factor-covered if every edge of \(G\) is contained in some \(f\)-factor. \(G\) is called \(f\)-factor-deleted if \(G\) – \(e\) contains an \(f\)-factor for every edge \(e\). Babler proved that every \(r\)-regular, \((r – 1)\)-edge-connected graph of even order has a \(1\)-factor. In the present article, we prove that every \(2r\)-regular graph of odd order is both \(2m\)-factor-covered and \(2m\)-factor-deleted for all integers \(m\), \(1 \leq m \leq r – 1\), and every \(r\)-regular, \((r – 1)\)-edge-connected graph of even order is both \(m\)-factor-covered and \(m\)-factor-deleted for all integers \(m\), \(1 \leq m \leq \left\lfloor \frac{r}{2} \right\rfloor\).