# Intuitive reason behind the fact that the definition of bipartite graph assumes a simple graph

I was going through the text "Discrete Mathematics and its Applications" by Kenneth Rosen where I came across the definition of bipartite graph, which assumes a the graph to be simple.

A simple graph $$G$$ is called bipartite if its vertex set $$V$$ can be partitioned into two disjoint sets $$V_1$$ and $$V_2$$ such that every edge in the graph connects a vertex in $$V_1$$ and a vertex in $$V_2$$ (so that no edge in G connects either two vertices in $$V_1$$ or two vertices in $$V_2$$). When this condition holds, we call the pair $$(V_1,V_2)$$ a bipartition of the vertex set $$V$$ of $$G$$.

Now from the condition that each edge should run only between two bipartitions so formed, we cannot have a self loop as it an edge from a vertex to itself in the same bipartition. But what restrictions do we have to include parallel edges running between the two bipartitions? I hope there is none. So we can have multigraphs as bipartite without violating the terms conveyed by the meaning of the definition.

• I don’t see any obstruction to defining a notion of bipartite multigraph or bipartite directed graph. But I have seen bipartite graphs only in the context of the theory of simple graphs... Jun 7, 2020 at 17:04
• @PrudiiArca: You can find bipartite digraphs fairly easily if you go looking for them; e.g., in Jørgen Bang-Jensen & Gregory Gutin, Digraphs: Theory, Algorithms and Applications [772 page PDF] or this paper. Jun 7, 2020 at 23:39

Of course, the definition of "bipartite" is easily generalised to graphs that are not simple, and we might want to do this in some cases: for instance if we are studying graph colourability, we might want to use "bipartite" as synonymous with "$$2$$-colourable".
Since in the text it is phrased as assuming that $$G$$ is bipartite, it says nothing about whether or not other graphs can be bipartite, so I would assume they only care about bipartite graphs in the context of simple graphs. To summarise there is no explicit obstruction to defining bipartite for non-simple graphs and it's usually clear from context whether you want to or not.