Subcritical component bound for a binomial random graph

For every fixed $\epsilon >0$, every component of $G(n,(1-\epsilon )/n)$ has $O_{\epsilon }(\log n)$ vertices with high probability. A breadth-first exploration is dominated by a branching process of mean $1-\epsilon$; its total progeny has an exponentially decreasing tail.

A connected graph is unicyclic when it contains exactly one cycle, equivalently when its numbers of vertices and edges agree.