$*(4)$ Let n $=$ v$($G$),$ m $=$ e$($G$)$ be the vertex number and edge number of

G$,$ respectively.

$($a$)$ If m $>=$ n$,$ then G contains a cycle.

$($b$)$ If m $>=$ n $+4,$ then G contains two edge$-$disjoint cycles.

Hint for item $(2).$ We will induct on n$.$ We can check by hand the

conclusion for n $<=4,$ so we put the base case n $=4.$ Then we can discuss

the cases when $\backslash$delta $($G$)=1$ or $\backslash$delta $($G$)=2.$