# 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?

## Related Questions

### Multiplying and adding fractions

4  Asked on November 26, 2021

### Bijection between tensors and permutations (in linear $O(n)$ time)

1  Asked on November 26, 2021 by nikos-m

### How to Solve a Stars and Bars Discrete Math Problem

2  Asked on November 26, 2021 by kristen-m-day

### Closed form of $int_0^infty arctan^2 left (frac{2x}{1 + x^2} right ) , dx$

3  Asked on November 26, 2021

### Finding the limit of $mathbb{E}[theta^n]/mathbb{E}[theta^{n-1}]$

1  Asked on November 26, 2021

### Degree of a determinant

1  Asked on November 26, 2021

### Are basic feasible solutions, vertices, and extreme points equivalent for semidefinite programs (SDPs)?

1  Asked on November 26, 2021

### Find all functions $f:mathbb{R}^+to mathbb{R}$ such that $xf(xf(x)-4)-1=4x$

2  Asked on November 26, 2021

### Fourier transform of $1/ sqrt{m^2+p_1^2+p_2^2+p_3^2}$

1  Asked on November 26, 2021 by sebastien-b

### Optimisation of norm of matrices without the elements on diagonals

0  Asked on November 26, 2021 by nikowielopolski

### Given $frac{z_1}{2z_2}+frac{2z_2}{z_1} = i$ and $0, z_1, z_2$ form two triangles with $A, B$ the least angles of each. Find $cot A +cot B$

2  Asked on November 26, 2021 by seo

### Notation for “and”

1  Asked on November 26, 2021

### Proving that, given a surjective linear map, a set is a generator of a vector space

1  Asked on November 26, 2021 by jd_pm

### Production function problem (Lagrange multiplier)

1  Asked on November 26, 2021

### Brezis-Kato regularity argument – Some questions about Struwe’s proof Part II

1  Asked on November 26, 2021 by danilo-gregorin-afonso

### Understanding the multidimensional chain rule

1  Asked on November 26, 2021 by atw

### Prove identity matrix with singular value decomposition

1  Asked on November 26, 2021 by hojas

### Number of possible n-towers

1  Asked on November 26, 2021 by aatish-five

### Convergence of $sqrt{1-sqrt{{1}-sqrt{{1}-sqrt{{1-sqrt{{1}}}…}}}}$

1  Asked on November 24, 2021

### Is this series $sum_{n=0}^{infty} 2^{(-1)^n – n} = 3$ or $approx 3$

2  Asked on November 24, 2021

### Ask a Question

Get help from others!