Theorem. Every D
2
C(G) is a disjoint union of circuits.
Why this is hard: For all we know, D might be the sum of finite face bdries in the
wild-circle graph. (Ie, D is the wild circuit, but we don‘t know that from the way its given as
an infinite thin sum.) For finite G, we‘d use that D induces even degrees everywhere, and then
greedily move along from the given edge e until we find a cycle in D through e. Here, we also
know that all vx degrees are even, but moving along gets us nowhere.
Theorem. Every D
2
C(G) is a disjoint union of circuits.
Proof. Enumerate the edges in D, and greedily find in the set D0 2 C of
remaining edges a circuit C through the smallest edge e = xy 2 D0 . Delete
E(C), and repeat.
Theorem. Every D
2
C(G) is a disjoint union of circuits.
Proof. Enumerate the edges in D, and greedily find in the set D0 2 C of
remaining edges a circuit C through the smallest edge e = xy 2 D0 . Delete
E(C), and repeat.
To find C, we construct an x–y arc A in the standard subspace D0 r {e}. We
obtain A in (not as) the limit of x–y arcs An in Gn .
Theorem. Every D
2
C(G) is a disjoint union of circuits.
Proof. Enumerate the edges in D, and greedily find in the set D0 2 C of
remaining edges a circuit C through the smallest edge e = xy 2 D0 . Delete
E(C), and repeat.
To find C, we construct an x–y arc A in the standard subspace D0 r {e}. We
obtain A in (not as) the limit of x–y arcs An in Gn .
D0 \ E(Gn ) 2 C(Gn ) by the cut criterion. (Cuts of Gn are cuts of G.)
Pick a circuit Cn 3 e, and put An := Cn r {e}.
cut criterion: meets every cut evenly
pick a circuit: by finite decomposition theorem
(all for finite graphs)
Theorem. Every D
2
C(G) is a disjoint union of circuits.
Proof. Enumerate the edges in D, and greedily find in the set D0 2 C of
remaining edges a circuit C through the smallest edge e = xy 2 D0 . Delete
E(C), and repeat.
To find C, we construct an x–y arc A in the standard subspace D0 r {e}. We
obtain A in (not as) the limit of x–y arcs An in Gn .
D0 \ E(Gn ) 2 C(Gn ) by the cut criterion. (Cuts of Gn are cuts of G.)
Pick a circuit Cn 3 e, and put An := Cn r {e}.
Now A exists by Lemma P (‘method 2’).
⇤
This was the hotchpotch lemma, that there‘s an x–y arc in a standard subspace X ✓ |G| if
for all large enough n there‘s a path from n (x) to n (y) in Gn , where n is the projection
|G| ! Gn (which maps all ends and some vertices to dummy vertices).
Theorem. Every D
2
C(G) is a disjoint union of circuits.
Proof. Enumerate the edges in D, and greedily find in the set D0 2 C of
remaining edges a circuit C through the smallest edge e = xy 2 D0 . Delete
E(C), and repeat.
To find C, we construct an x–y arc A in the standard subspace D0 r {e}. We
obtain A in (not as) the limit of x–y arcs An in Gn .
D0 \ E(Gn ) 2 C(Gn ) by the cut criterion. (Cuts of Gn are cuts of G.)
Pick a circuit Cn 3 e, and put An := Cn r {e}. (We’ll just use ‘An connected’).
Else, for 1-lemma:
Vn := {Am \ E(Gn ) | m > n} .
As Gn is contracted from Gm , these Am \ E(Gn ) are connected in Gn .
S
Ray D0 ✓ D1 ✓ . . . ! find x–y arc A in X := n Dn (63 e).
As X is standard, it suffices to show that X is connected, ie., has an edge in
every finite cut F such that X meets both sides of F (wlog in vertices u, v 2 X);
cf. Lemma 8.6.5.
Choose n large enough that Gn contains u, v, F . Then F is also a cut of Gn
(components of contracted v’s each lie on one side of F ). Since Dn = Am \ E(Gn )
is connected and contains u, v, we have
; 6= Dn \ F = E(X) \ F
as desired.
⇤
Theorem 2.5. [Survey ‘Locally finite graphs with ends’]
The following statements are equivalent for sets D ✓ E(G):
(i) D
2
C(G), that is to say, D is a sum of circuits in |G|.
(ii) Every component of D admits a topological Euler tour.
(iii) Every vertex and every end has even (edge-) degree in D.
(iv) D meets every finite cut in an even number of edges.
Remarks:
(i) $ (iv): shown last week;
(ii) $ (iv): was an exercise for finding arcs by ‘method 2’.
(iii) ?
Aim of (iii) & (iv): combinatorial characterization of C
(iii) Even degrees at vertices are not enough:
All vertices have even degree. The (entire) edge set of the left graph
lies in C, but that of the right graph does not (by (i) $ (iv)).
The edge set on the left is a disjoint
union of squares, hence in C.
(iii) Even degrees at vertices are not enough:
All vertices have even degree. The (entire) edge set of the left graph
lies in C, but that of the right graph does not (by (i) $ (iv)).
Need : def of parity for infinite (edge-) degrees of ends in standard subspaces
edge-degrees: this is historical. Edge-cuts matter, but no-one has thought about whether
end degrees in terms of (entirely) disjoint arcs can work too.
In order for (i)$(iii) to become true we need that in the left graph the end has even degree in
X = |G|, but the two ends of the right graph have odd degree in X = |G|.
The right graph shows that we cannot call all infinite end-degrees even‘,
’
while the left graph shows that we cannot call them all odd‘.
’
So we
need to divide ends of infinite degree into ’even‘ and ’odd‘.
Definition 1: The edge-degree of an end ! of G in a standard subspace X ✓ |G|
is even if for every small enough neighbourhood U of ! the maximum number
of edge-disjoint arcs in X from X r U to ! is even.
ASK: what does every small enough mean?
Tafel: there 9 a (possibly small) nbhd W such that 8 nbhd U ✓ W . . . (‘eventually’)
This is topological!
Aim of (iii): combinatorial characterization of C
Definition 1: The edge-degree of an end ! of G in a standard subspace X ✓ |G|
is even if for every small enough neighbourhood U of ! the maximum number
of edge-disjoint arcs in X from X r U to ! is even.
Combinatorial equivalent:
9 S0 : 8 S ◆ S0 : minS 0 ◆S |E(S 0 , !) \ E(X)| is even
(with S0 , S, S 0 finite and E(S 0 , !) := edges from S 0 to C(S 0 , !)).
Ĉ(S, !) is the small enough nbhd‘ from Def. 1: the fact that S ◆ S0 ensures small enough‘,
’
’
and the minS 0 ◆S |E(S 0 , !) \ E(X)| determines the max. number of edge-disjoint arcs
in X from S to !.
Perhaps set as Ü (assuming the arc-building exercises set earlier)
Definition 2: The degree of an end ! of G in a standard subspace X ✓ |G|
is weakly even if ! has arbitrarily small neighbourhoods U in X for which the
maximum number of edge-disjoint arcs in X from X r U to ! is even.
Tafel: there 8nbhd W
9 nbhd U
✓ W ...
(‘again and again’)
Definition 2: The degree of an end ! of G in a standard subspace X ✓ |G|
is weakly even if ! has arbitrarily small neighbourhoods U in X for which the
maximum number of edge-disjoint arcs in X from X r U to ! is even.
Combinatorial equivalent:
0
8 S0 : 9 S ◆ S0 : min
|E(S
, !) \ E(X)| is even
0
S ◆S
(with S0 , S, S 0 finite and E(S 0 , !) := edges from S 0 to C(S 0 , !)).
Definition 1: The edge-degree of an end ! of G in a standard subspace X ✓ |G|
is even if for every small enough neighbourhood U of ! the maximum number
of edge-disjoint arcs in X from X r U to ! is even.
Definition 2: The degree of an end ! of G in a standard subspace X ✓ |G|
is weakly even if ! has arbitrarily small neighbourhoods U in X for which the
maximum number of edge-disjoint arcs in X from X r U to ! is even.
Theorem. (Berger & Bruhn, 2011)
Definition 1 makes Theorem 2.5 true.
Definition 1: The edge-degree of an end ! of G in a standard subspace X ✓ |G|
is even if for every small enough neighbourhood U of ! the maximum number
of edge-disjoint arcs in X from X r U to ! is even.
Definition 2: The degree of an end ! of G in a standard subspace X ✓ |G|
is weakly even if ! has arbitrarily small neighbourhoods U in X for which the
maximum number of edge-disjoint arcs in X from X r U to ! is even.
Theorem. (Berger & Bruhn, 2011)
Definition 1 makes Theorem 2.5 true.
Conjecture. (Berger & Bruhn, 2011)
Definition 2 makes Theorem 2.5 true.
Definition 1: The edge-degree of an end ! of G in a standard subspace X ✓ |G|
is even if for every small enough neighbourhood U of ! the maximum number
of edge-disjoint arcs in X from X r U to ! is even.
Definition 2: The degree of an end ! of G in a standard subspace X ✓ |G|
is weakly even if ! has arbitrarily small neighbourhoods U in X for which the
maximum number of edge-disjoint arcs in X from X r U to ! is even.
Theorem. (Berger & Bruhn, 2011)
Definition 1 makes Theorem 2.5 true.
Conjecture. (Berger & Bruhn, 2011)
Definition 2 makes Theorem 2.5 true.
Note: As such, Definitions 1 and 2 are not equivalent.
If we replace in a ray the
resulting graph has both
nth edge with n parallel edges, for every n 2 N, the end of the
weakly even and odd degree.
But, unlike the X in the Thm, this example has vertices of odd degree. If there are no such
examples in which all vertices have even degree, the Conjecture will be true.
Orthogonality
Call two sets D, F ✓ E(G) orthogonal if |D \ F | is finite and even.
We‘re not assuming that D \ F is finite but requiring it.
If |D \ F | = 1, then D, F are not orthogonal (rather than this being undef’d).
Orthogonality
Call two sets D, F ✓ E(G) orthogonal if |D \ F | is finite and even.
F ? := { D ✓ E(G) | D is orthogonal to every F
2
F }.
Orthogonality
Call two sets D, F ✓ E(G) orthogonal if |D \ F | is finite and even.
F ? := { D ✓ E(G) | D is orthogonal to every F
2
F }.
?
Write Ffin for the set of finite elements of F, and (Ffin )? =: Ffin
.
So Bfin is just the set of finite cuts.
But Cfin is the set of finite edge sets D that are sums, maybe infinite sums, of circuits (maybe
infinite circuits). However, since D is also a disjoint union of circuits, which must be finite if
D if finite, Cfin is also the space of finite sums of finite circuits. This is a minus-excercise.
Orthogonality
Call two sets D, F ✓ E(G) orthogonal if |D \ F | is finite and even.
F ? := { D ✓ E(G) | D is orthogonal to every F
2
F }.
Write Ffin for the set of finite elements of F.
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
Remember as: All equations containing each of =, C, B, fin , ? exactly once.
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
Proof. (i) was shown before.
It’s Lemma 2.2 (i) as well as (i)$(iv) of Thm.2.5, but was shown earlier in our proof of
Thm. 2.1 (ii).
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
Proof. (ii) ‘◆’ (the ‘hard’ direction): by our Lemma 2.2
We proved not only that every set which is orthogonal to every finite element of C is a cut, but
‘even’ that every set which is (‘only’) orthogonal to every finite circuit is a cut. But, of course,
a set orthogonal to every finite circuit is also orthogonal to every finite D 2 C, since D is a
disjoint union of (finite) circuits.
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
Proof. (ii) ‘◆’ (the ‘hard’ direction): by our Lemma 2.2
‘✓’ (the ‘easy’ direction):
By the jumping arc lemma, every cut F is orthogonal to every finite circuit. But why should
this imply that it‘s orthogonal to every thin sum D 2 C of circuits that happens to
come out finite? Answer: because D is also a disjoint union of circuits, and these must
be finite if D is finite.
Before explaining this, ask why the following isn‘t a counterexample:
Consider, eg, as F the rungs of the single ladder, and as D 2 Cfin the sum of squares.
This sum meets F only in the first rung.
(This isn‘t a counterexample, because the sum of the squares is an infinite edge set (it contains
both sides of the ladder); it only meets F finitely.)
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
Proof. (ii) ‘◆’ (the ‘hard’ direction): by our Lemma 2.2
‘✓’ (the ‘easy’ direction): by the C-decomposition theorem.
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iii) C ? ◆ Bfin is equivalent to C ✓ Bfin
shown under (i).
Both say:
Whenever we take a finite cut an an element of C,. . .
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iii) C ? ◆ Bfin is equivalent to C ✓ Bfin
shown under (i).
‘✓’: Every F
If F meets every D
2
2
C ? is a cut, by ◆ of (ii).
C evenly, then in particular every D
2
Cfin .
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iii) C ? ◆ Bfin is equivalent to C ✓ Bfin
shown under (i).
‘✓’: Every F
2
Yes, if G is an infinite
but is not a finite cut.
C ? is a cut, by ◆ of (ii). Can it be infinite?
tree, then F
Amazing
Assuming 2-edge-connected, we‘ll
it infinitely.
= E(G) meets every circuit evenly (since there is none)
that assuming ‘2-edge-connected’ can mend this.
construct for any infinite cut some D
2
C meeting
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iii) C ? ◆ Bfin is equivalent to C ✓ Bfin
shown under (i).
‘✓’: Every F
2
C ? is a cut, by ◆ of (ii). Can it be infinite?
If |F | = 1, pick 8 f
2
F a finite circuit C(f ) through f .
Exists, since by 2-edge-connected, the edge f does
any path linking them makes a circuit with f .
not disconnect its endv‘s;
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iii) C ? ◆ Bfin is equivalent to C ✓ Bfin
shown under (i).
‘✓’: Every F
2
C ? is a cut, by ◆ of (ii). Can it be infinite?
If |F | = 1, pick 8 f
2
F a finite circuit C(f ) through f .
Pick an NST T . Its fundamental circuits generate C and are finite, so wlog the
C(f ) are fundamental circuits of T .
Just generate C(f ) from fund.circuits: one of them must contribute f .
The C(f ) may coincide for di↵erent f , and f may lie on T .
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iii) C ? ◆ Bfin is equivalent to C ✓ Bfin
shown under (i).
‘✓’: Every F
2
C ? is a cut, by ◆ of (ii). Can it be infinite?
If |F | = 1, pick 8 f
2
F a finite circuit C(f ) through f .
Pick an NST T . Its fundamental circuits generate C and are finite, so wlog the
C(f ) are fundamental circuits of T .
Inductively, select 1’ly many disjoint C(f ) (possible, as they form a thin family).
Their sum D 2 C meets F infinitely, not evenly, so F 2/ C ? (contradiction).
Each fundamental circuit of T is finite, and hence is picked by only finitely many f (since the f
are distinct). So the set of all C(f ) ever picked is infinite. Each of them meets only finitely
many others, because every edge of G (especially, of T ) lies in only finitely many fund.circuits,
since the fund.cuts of T are finite.
ASK: Could we have used any ordinary sp.tree? Or any TST?
No: T must be both an ordinary sp.tree (so that fund.circuits are finite) and a TST, because
we need that its fund.circuits form a thin family (so each meets only finitely many others).
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iv) B? ◆ Cfin is equivalent to B ✓ Cfin
shown under (ii).
Whenever. . .
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iv) B? ◆ Cfin is equivalent to B ✓ Cfin
shown under (ii).
‘✓’: Every D
2
B? lies in C, by ◆ of (i). Can it be infinite?
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
?
Proof. (iv) B? ◆ Cfin is equivalent to B ✓ Cfin
shown under (ii).
‘✓’: Every D
2
B? lies in C, by ◆ of (i). Can it be infinite?
No: if |D| = 1, we can pick an infinite independent subset and extend this to a
cut meeting D infinitely rather than evenly.
⇤
We can pick 1’ly many independent edges inductively, because G is locally finite. For each of
these edges, put one endvertex in a set A, the other in B. Extend the disjoint union A [ B
arbitrarily to a bipartition of V (G). The cut it defines contains those infinitely many edges.
Alternatively, one could dualise the proof of (iii): every edge of our infinite D lies in a
finite cut (eg, the atomic bond of an endvertex of it), the fundamental cuts of an NST generate
these, so each edge of D lies in a (finite) fundamental cut, and we can find infinitely many
among them that are disjoint on edges of D. Their sum, then, is a cut meeting D infinitely.
Puzzle: the direct proof is so much easier than the dualised proof of (iii). Can we, conversely,
dualise the proof of (iv) using vx partitions to obtain a simpler proof of (iii)?
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
The consideration of C and B in graphs comes from homology: Cfin is the group
of 1-chains with vanishing boundary (finite formal sums of edges inducing even
degrees at all the vertices), B is the group of 0-coboundaries (maps E ! Z2
determined by evaluating a given map V ! Z2 on the edge boundaries), and
Theorem 2.6 has topological analogues in higher dimensions.
Assign talk on Poincare duality?
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
In graphic matroids, where we have no vertices but only edges, the sets of
minimal non-empty elements of C and B still play a role:
Cˆ := the set of circuits
B̂ := the set of bonds
Is there a result similar to Theorem 2.6 for these sets?
in graphs, not in matroids - though one can ask that too
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
In graphic matroids, where we have no vertices but only edges, the sets of
minimal non-empty elements of C and B still play a role:
Cˆ := the set of circuits
B̂ := the set of bonds
Is there a result similar to Theorem 2.6 for these sets?
?
?
?
?
None analogous to (i) or (ii), since B̂fin
= Bfin
and Cˆfin
= Cfin
(but Cˆ 6= C and B̂ 6= B for all but a few graphs). But there is one for (iii)–(iv):
? too, because
Requires decomposition thm: a set orthogonal to all finite circuits is in Cfin
every finite element of C decomposes disjointly into circuits, which happen to be finite
(by disjointness), and orthogonality is preserved in finite sums (disjoint or not). Similarly for
bonds ! cuts.
Theorem 2.6. [Proof: see ‘Locally finite graphs with ends’]
Let C := C(G) and B := B(G).
?
(i) C = Bfin
.
?
(ii) B = Cfin
.
(iii) If G is 2-edge-connected, then C ? = Bfin .
(iv) B? = Cfin .
In graphic matroids, where we have no vertices but only edges, the sets of
minimal non-empty elements of C and B still play a role:
Cˆ := the set of circuits
B̂ := the set of bonds
Theorem 2.6 0 . (Diestel & Pott; 2014)
Cˆ? = Bfin = C ?
and
B̂? = Cfin = B?
when G is 3-connected.
Analogues of (iii) and (iv) for orthogonality to just circuits and bonds.
Fails for just 2-connected; that‘s an exercise.