Problem $3\mathrm{.}7($a$).($Additional proof for part $($a$).)$ A bipartite graph is a graph G $=($V$,$ E$)$ whose vertices can be partitioned into two sets $($V $=$ V$1\backslash$cup V$2$ and V$1\backslash$cap V$2=)$ such that there are not edges between vertices in the same set $($for instance, if u$,$ v in V$1,$ then there is no edge between u and v$).$

$($a$)$ Give a linear$-$time algorithm to determine whether an undirected graph is bipartite. Show your algorithm$$s runtime using big$-$O notation.