A Neighborhood Union Condition for Fractional \((k,m)\)-Deleted Graphs

Abstract

A graph \(G\) is called a fractional \((k, m)\)-deleted graph if any \(m\) edges are removed from \(G\), then the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for integers \(k \geq 2\), \(m \geq 0\), \(n \geq 8k + 4m – 7\), and \(\delta(G) \geq k + m\), if
\[|N_G(x) \cup N_G(y)| \geq \frac{n}{2}\]
for each pair of non-adjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \((k, m)\)-deleted graph. The bounds for neighborhood union condition, order, and the minimum degree of \(G\) are all sharp.