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

Question

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.

Answer 1

Whether we want to allow non-simple graphs to be bipartite or not is heavily dependent on context.

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".

However, other times, it does make sense to restrict ourselves to simple graphs. When trying to determine maximum matchings for instance, then we usually want to talk about a single edge capacity between two nodes, so it is not useful to allow non-simple graphs to be bipartite in this context. (Indeed, a non-simple graph can trivially be reduced to a simple graph for the purposes of maximum matching.)

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.