Can Bipartite graph contain parallel edges or self-loops?

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I have some confusion about definition of the bipartite graph. The definition of bipartite graph doesn't mention about multiple edges between two vertices. Can a graph with parallel edges become a bipartite graph if their vertices are partitioned into V1 and V2?

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Usually no, but self-loops and parallel edges are usually forbidden for non-bipartite graphs as well. It will depend on context. Probably if your book doesn't say, it means to exclude those cases. But it's hard to be completely sure without seeing what it says.

However, note that no bipartite graph has self-loops, even if they are not expressly forbidden.