Sufficient conditions for fractional k-extendable graphs

Abstract

Let $k$ and $n$ be two positive integers. A graph $G$ is said to be fractional $k$-extendable for $0\leq k\leq\frac{n-2}{2}$ if every
$k$-matching $M$ in $G$ is contained in a fractional perfect matching $G[F_h]$ of $G$ such that $h(e)=1$ for all $e\in M$, where $h:E(G)\rightarrow[0,1]$ be a function. Let $e(G)$ denote the size of $G$ and $\rho(G)$ denote the spectral radius of $G$. In this paper, we first provide a tight size condition to ensure that a connected graph is fractional $k$-extendable. Then, we determine a lower bound on the spectral radius of a connected graph $G$ to guarantee that $G$ is fractional $k$-extendable. Finally, we construct some extremal graphs to show that all the bounds are sharp.