Should the subsets in this graph theory exercise be finite?

Mathematics Asked by laurencevs on January 3, 2022

I’m reading Béla Bollobás’s Modern Graph Theory and one of the exercises (I.10) says the following:

Show that in an infinite graph $G$ with countably many edges there exists a set of cycles and two-way infinite paths such that each edge of $G$ belongs to exactly one of these iff for every $X subset V(G)$ either there are infinitely many edges joining $X$ to $V(G)-X$, or else $e(X,V(G)-X)$ is even.

I’m a bit confused by this, because if we consider a single infinite path $dots, -2, -1, 0, 1, 2, dots$ and take $X=lbrace1,2,3,dotsrbrace$ then I think we find that $e(X,V(G)-X)$ is one: there is only the edge from $0$ to $1$. Thus this graph satisfies the first requirement but not the second, contradicting the claim that they are equivalent.

Should the statement specify that the sets $X$ are to be finite, or have I missed something here?

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