Every tree is a bipartite. Can we show, if it is true, that it has only one possible "split" or in other words only 2 possible symmetric vertex colorings (red-blue) ? Is it true for any bipartite graph?
2026-03-30 06:19:59.1774851599
Different colorings of bipartite graph
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This is true for any (non-empty) connected bipartite graph, or more generally, the number of bicolorings is $2^k$ where $k$ is the number of connected components.
The proof is rather simple: For each connected component you distinguish one vertex which determines the coloring. Once you color the distinguished vertex (two choices), everything else in its connected component is determined.
I hope this helps $\ddot\smile$