I was just wondering what the correlation is between a breadth-first search tree of a graph and that graph being bipartite?
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A bipartite graph is a graph that you can $2$-color. If you fix the color of one vertex (the source), then the rest of the coloring (provided that it exists) is totally defined by the distance to the source vertex. Hence, your graph is bipartite if and only if there is no "horizontal" edge in your BFS tree.