Question

# A closed path is a path from v to v. Show that a connected graph G is bipartite if and only if every closed path in G has even length.

Solution

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Given: A $\textbf{closed path}$ is a path from $v$ to $v$

A $\textbf{bipartite graph}$ is a simple graph whose vertices can be partitioned into two sets $V_1$ and $V_2$ such that there are no edges among the vertices of $V_1$ and no edges among the vertices of $V_2$, while there can be edges between a vertex of $V_1$ and a vertex of $V_2$.

To proof: A connected graph $G$ is bipartite if and only if every closed path in $G$ has even length.

$\textbf{PROOF}$

We will prove that "A connected graph $G$ is bipartite if and only if every closed path in $G$ has even length" by showing that every closed path in $G$ has even length when a connected graph $G$ is bipartite and showing that a connected graph $G$ is bipartite when every closed path in $G$ has even length.

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