I want to prove that every bipartite graph has an equitable edge-bicoloring (not necessarily proper). That is, I want to find an edge-coloring with two colors, say red and blue, such that |# red edges - # blue edges| < 2 for each vertex.
I was trying to use induction on the number of edges. If there is a vertex with odd degree, then we can delete any edge incident to the vertex and apply the induction hypothesis. However, I have no idea when every vertex has even degree. Thank you.
If every vertex has even degree and the graph is not empty, then the graph has a cycle. Remove this cycle and apply induction.