Can every graph have an edge cut $[X,Y]$ such that $G[X]$ and $G[Y]$ are even?

Asked Oct 28 '23 at 13:20

Active Oct 28 '23 at 13:20

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I am stumped by an exercise in Bondy’s book:

Show that every graph has an edge cut $[X,Y]$ such that $G[X]$ and $G[Y]$ are even.

I tried many ideas, such as making induction on the number of edges, considering how to adjust the old edge cut after removing an edge and then putting it back, or taking out a circle and then putting it back, but without exception, there were always uncontrollable vertices that could not be discussed.. Could you please give me some answers or tips? Thanks a lot!:)

asked Oct 28 '23 at 13:20

Jungang Chen

Does this answer your question? The details may depend on whether you require $X$ and $Y$ to be nonempty.. – Izaak van Dongen Oct 28 '23 at 14:54
1

@IzaakvanDongen This is exactly what I want! Clever construction!Thanks a lot! – Jungang Chen Oct 29 '23 at 07:27

Can every graph have an edge cut $[X,Y]$ such that $G[X]$ and $G[Y]$ are even?

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