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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