May a bipartite graph have self-loops?

Question

By definition I think no, but I'm not so sure.

Thanks in advance

Answer 1

Say that the graph with vertices $V$ is bipartite, hence $V=A \cup B$ and $A \cap B = \emptyset$ where all edges connect one vertex in $A$ with one in $B$. Assume there is an element with a "self-loop":

Pick one of those elements, i.e. a node with an edge that connects back to itself, for example $a_0 \in A$.

Because the graph is bipartite, all edges that connect $a_0$ only connect "at the other end" with $b_i \in B$. But we must have $b_i \not= a_0$ for all $i$ since $A \cap B = \emptyset$.

Hence: No, a bipartite graph cannot have an element with a self-loop.

Answer 2

Presumably, a self-loop is an edge that starts and ends at the same vertex. If so, then you're right: no graph with a "self-loop" is bipartite.