Degree Conditions for Graphs to Be Fractional \(k\)-Covered Graphs

Abstract

Let \(k \geq 3\) be an integer, and let \(G\) be a graph of order \(n\) with \(n \geq \max\{10, 4k-3\}\) and \(\delta(G) \geq k+1\). If \(G\) satisfies \(\max\{d_G(x), d_G(y)\} \geq \frac{n}{2}\) for each pair of nonadjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \(k\)-covered graph. The result is best possible in some sense, and it improves and extends the result of C. Wang and C. Ji (C. Wang and C. Ji, Some new results on \(k\)-covered graphs, Mathematica Applicata \(11(1) (1998), 61-64)\).