Let $n \geq 2$ be an integer. What is the smallest positive integer for which every graph $G$ of order $n$ and size $m$ contains a 1-factor?
My work: I know that $m \geq \binom{n-1}{2}+1$ as if otherwise you could just form the graph $$G=K_{n-1} \cup K_1$$ which will not contain a 1-factor for all $n$. Now where i'm confused is showing that all graphs of size at least $\binom{n-1}{2}+1$ actually contain a 1-factor. If anyone could provide a hint to get me started that would be greatly appreciated!