Degree Conditions for Fractional \((k,m)\)-Deleted Graphs

Abstract

A graph \(G\) is called a fractional \((k, m)\)-deleted graph if after deleting any \(m\) edges of \(G\), the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for \(k \geq 2\) and \(m \geq 0\), \(G\) is a fractional \((k, m)\)-deleted graph if one of the following conditions holds: 1) \(n \geq 4k + 4m – 3\), \(\delta(G) \geq k + m\), and \(\max\{d_G(u), d_G(v)\} \geq \frac{n}{2}\) for each pair of non-adjacent vertices \(u\) and \(v\) of \(G\); 2) \(\delta(G) \geq k + m\), \(\omega_2(G) \geq n\), \(n \geq 4k + 4m – 5\) if \((k, m) = (3, 0)\), and \(n \geq 8\) if \((k, m) = (3, 0)\). The results are best possible in some sense.