$4.(10$ pts $)$ Consider an undirected graph with $\backslash ($ n $\backslash )$ vertices $($assume $\backslash ($ n $\backslash )$ is even$),$ no self loops, and each vertex has degree $\backslash (\backslash$geq $\backslash$frac$\{$n$\}\{2\}\backslash )($degree of a vertex is the number of edges incident at a vertex$).$ Prove or disprove that all vertices in the graph belong to one connected component.