A bipartite graph is a graph $G$ whose vertices can be 2-colored, i.e., such that there exists a graph homomorphism $G \to K_2$. Note that a bipartite graph cannot have any loops, although it may have multiple edges.
R. Diestel, Graph Theory , GTM 173 Springer Heidelberg 2000².
F. W. Lawvere, Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs , Revista Colombiana de Matemáticas XX 1986 pp.179-186. Reprinted with an author comment as TAC Reprint no.9 (2005) pp.1-7. (pdf)