So i have this task i am a little bit lost how to approach this problem.
Could anyone guide me trough it?
Let G be a connected graph with at least one cycle. Proof that G is bipartite when no cycle is odd.
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Start with two empty sets $X$ and $Y$. Pick any vertex $v_0$ and put it in $X$. Then put in $X$ any vertex that is connected to $v_0$ by a path of even length and put in $Y$ those at an odd length. To see why this is a well defined function, observe that if a vertex is connected to $v_0$ by both a path of even and of odd lengths, then that would produce a cycle of odd length. Observe also that there cannot be edges between elements of $X$, since that would produce paths of odd length to $v_0$, and no edge between elements of $Y$ either. – plop Jun 17 '21 at 14:51
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Howdy: https://math.stackexchange.com/questions/311665/proof-a-graph-is-bipartite-if-and-only-if-it-contains-no-odd-cycles – Asinomás Jun 17 '21 at 14:54