I'm having a bit of trouble with this homework problem: If $G=(V, E)$ is a connected bipartite undirected graph with $V$ partitioned as $V_1\cup V_2$, how can I prove the existence of a Hamilton path in $G$ implies that $|V_1|-|V_2|\in\{-1, 0, 1\}$?
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Hint: Since $G$ is bipartite, any two adjacent vertices in your Hamiltonian path must be in separate subsets. How many choices are there for the subset that contains the initial vertex of the path? How many choices are there for the subset that contains the terminal vertex of the path? What do these cases together imply about the cardinalities of $V_1$ and $V_2$? |
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