Augment depth-first search with a two-valued colour:
paint(v, c):
    colour[v] = c
    for w in adjacency[v]:
        if colour[w] is unset:
            if not paint(w, 1-c):
                return false
        else if colour[w] == c:
            return false
    return true

for v in V:
    if colour[v] is unset:
        if not paint(v, 0):
            return false
return true
Each tree edge receives opposite endpoint colours. If an inspected edge has equal colours, the preceding parity argument produces an odd cycle, so the graph is not bipartite. If the procedure succeeds, every edge has differently coloured endpoints, and the two colour classes are a bipartition. The outer loop handles disconnected graphs and isolated vertices.
Each vertex is coloured once and each edge incidence is checked once, giving time and extra space with adjacency lists. No separate cycle enumeration is required.

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