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# Is there a bipartite analogueanalog of graph theory?

I would like to compile a list of questions about graphs that have a non-trivial analogs for bipartite graphs.

Let me give the following examples:

1. Cycle vs Even cycle. Most questions about cycles in graphs have analogs in even cycles for bipartite graphs. For instance, it is trivial to show that a bipartite graph on an odd number of vertices cannot have a Hamilton cycle. In such a case the bipartite analog of a Hamilton cycle is a cycle missing exactly one vertex.

2. Minimal Girth. For graphs, 3 is the minimal possible length of a cycle. For bipartite graphs, the analogous number is 4.*

3. Triangular vs quadrangular embeddings. In topological graph theory, a triangular embedding of a simple graph determine its genus. For bipartite graphs, the analog is an embedding with quadrangles as faces.