Both BFS and DFS solve bipartite detection in time. Which should you use?
BFS advantages: iterative (no recursion stack limits), and natural level structure matches coloring. DFS advantages: often shorter code, and some find recursion more intuitive. For deep graphs (long chains), BFS is safer because recursion might hit stack limits. For typical problems, either works. Pick what you are comfortable with. Knowing both makes you flexible in interviews.
Space complexity is for the data structures used.