A survey on flows in graphs and matroids

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A survey on flows in graphs and matroids Bertrand Guenin Department of Combinatorics & Optimization, University of Waterloo, 200 University Ave. W, ON N2L3G1, Canada

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Article history: Received 10 December 2014 Received in revised form 16 October 2015 Accepted 26 October 2015 Available online xxxx Keywords: Flows Minimax relations Signed graphs Binary matroids

abstract A quintessential minimax relation is the Max-Flow Min-Cut Theorem which states that the largest amount of flow that can be sent between a pair of vertices in a graph is equal to the capacity of the smallest bottleneck separating these vertices. We survey generalizations of these results to multi-commodity flows and to flows in binary matroids. Two tantalizing conjectures by Seymour on the existence of fractional and integer flows are motivating our work. We will not assume from the reader any background in matroid theory. © 2015 Elsevier B.V. All rights reserved.

1. Flows in graphs In a flow problem we are given a graph G where the edges are partitioned into demand edges Σ and capacity edges E (G) − Σ .1 Every edge e is assigned a non-negative integer value we . For a demand edge e, we indicates the amount of flow required between the endpoints of e, and for a capacity edge e, we is the maximum amount of flow allowed on that edge. The triple (G, Σ , w) is a flow instance. An example of a flow instance is given in Fig. 1. For every circuit C that contains exactly one demand edge e = st we assign some non-negative value yC that indicates the amount of flow that is carried between s and t along the path C − {e}. (By a circuit we mean a set of edges that forms a connected graph where every vertex has degree two.) Denote by C the set of all circuits that contain exactly one demand edge. Then y ∈ ℜC is a flow if y ≥ 0 and the following constraints hold: Capacity constraints: for every edge e ∈ E (G) − Σ ,



(yC : e ∈ C ∈ C ) ≤ we ;

(1)

Demand constraints: for every edge e ∈ Σ ,



(yC : e ∈ C ∈ C ) = we .

(2)

Here (1) ensures that the total flow carried by a capacity edge e does not exceed its capacity we , and (2) guarantees that the total flow carried between the endpoints of a demand edge e is equal to its demand we . Consider the flow instance in Fig. 1. Set yC1 = 1

for C1 = {ac , cb, ba}

yC2 = 1

for C2 = {ac , cd, dg , gf , fa}

yC3 = 2

for C3 = {fd, dg , gf }

E-mail address: [email protected]. 1 Given sets A, B we denote by A − B the set {a ∈ A : a ̸∈ B}. http://dx.doi.org/10.1016/j.dam.2015.10.035 0166-218X/© 2015 Elsevier B.V. All rights reserved.

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Fig. 1. Flow instance: demand edges Σ are dashed, we is indicated on edge e.

Fig. 2. Flow instance: demand edges are dashed, for every edge e, we = 1.

and yC = 0 for all other circuits C ∈ C . It can be readily checked that the capacity and the demand constraints are satisfied for this instance. Thus y is a flow. Does every flow instance admit a flow? No. Indeed it can be readily checked that a necessary condition for the existence of a flow is that for every cut δ(U ) the total demand across the cut should not exceed the total capacity across the cut, in other words,

    w δ(U ) ∩ Σ ≤ w δ(U ) − Σ .2

(3)

This is known as the cut-condition. 1.1. Integer flows We say that flow y is integer if y is a vector with all integer entries. By a fractional flow we mean a flow y where y can take either integer or fractional values. Thus every integer flow is fractional but not vice-versa. A natural question is whether satisfying the cut-condition is sufficient to ensure the existence of an integer flow. The example in Fig. 2 shows that this is not the case. Here all the demand and capacity edges e are assigned the value we = 1. Note, that every cut contains at least as many solid edges as dashed edges, hence the cut-condition holds. Suppose, for a contradiction there exists an integer flow y. By symmetry we may assume that yC = 1 for C = {ab, bc , ca}. But then the capacity of both edges incident to c are fully used, and we cannot satisfy the demand for edge cd. Hence, there does not exist an integer flow. Theorem 1 states that if a flow instance does not ‘‘contain’’ that bad instance then the cut-condition is sufficient for the existence of an integer flow. Before we can formally state that result we need to define a containment operation. A signed graph is a pair (G, Σ ) where G is a graph and Σ is a subset of the edges. A set of edges B is odd if |B ∩ Σ | is odd and it is even otherwise. In particular, we will say that edges in Σ are odd. We define the following operations on a signed graph (G, Σ )3 :

• Deleting an edge e: replacing (G, Σ ) by (G\e, Σ − {e}); • Contracting an edge e ̸∈ Σ : replacing (G, Σ ) by (G/e, Σ ); • Resigning: replacing (G, Σ ) by (G, Σ △δ(U )) where δ(U ) is a cut of G.4 A signed graph (H , Γ ) is a minor of (G, Σ ) if it can be obtained from (G, Σ ) by repeatedly applying the aforementioned operations. An odd-K4 is the signed graph obtained from K4 where all edges are odd, i.e. it is (K4 , E (K4 )). Given a flow instance (G, Σ , w) we view (G, Σ ) as a signed graph where the demand edges are the odd edges. Observe that if we resign on the cut δ({a, b}) the bad instance in Fig. 2, we obtain odd-K4 . In particular, odd-K4 is a minor of the bad instance. We are now ready to state sufficient conditions for the existence of an integer flow (this a special case of Theorem 12 that will be described later).

2 For any S ⊆ E (G), w(S ) := e∈S we . 3 Given a graph H, then H /e (resp. H \e) is the graph obtained by contracting (resp. deleting) edge e. 4 For any sets A, B, A△B denotes the set A ∪ B − (A ∩ B).

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Fig. 3. Flow instance: demand edges are dashed, for every edge e, we = 1.

Theorem 1. Let (G, Σ , w) be a flow instance. If the cut-condition holds and (G, Σ ) has no odd-K4 minor then there exists an integer flow. There exists an algorithmic counterpart to this result, namely, Theorem 2. Let (G, Σ , w) be a flow instance. There exists an algorithm that will return, in polynomial time in the size of the flow instance, one of the following three outcomes: (a) an integer flow y, (b) a cut δ(U ) that violates the cut-condition, (c) an odd-K4 minor. In the case of outcome (c) the algorithm tells us what sequence of minor operations are required to get odd-K4 from (G, Σ ). This is a special case of a result of Truemper [27]. Thus in polynomial time we either find an integer flow, or find a reason why the flow does not exist, or find a certificate that indicates that we are outside the scope of Theorem 1. 1.2. Fractional flows Let us now turn our attention to fractional flows. Observe that the instance in Fig. 2 admits the following fractional flow, y{ab,bc ,ca} = y{ab,bd,da} = y{cd,da,ac } = y{cd,db,bc } =

1 2

.

Does this mean that the cut-condition is always sufficient for the existence of a fractional flow? The example in Fig. 3 shows that this is not the case. Here all the demand and capacity edges e are assigned the value we = 1. Note, that every cut contains at least as many solid edges as dashed edges, hence the cut-condition holds. However, we claim that there does not exist a fractional flow. Every circuit in C contains two capacity edges. It follows that to carry ϵ demand we require at least 2 × ϵ capacity. The total demand in our case is 4 which implies that we require 8 units of capacity. However, we only have 6 units of capacity available, hence no flow exists. An odd-K5 is the signed graph obtained from K5 where all edges are odd, i.e. it is (K5 , E (K5 )). Observe, that if we resign on the cut δ({a, b, c }) the bad instance in Fig. 3 we obtain odd-K5 . In particular, odd-K5 is a minor of that bad instance. We are now ready to state sufficient conditions for the existence of a fractional flow [13], Theorem 3. Let (G, Σ , w) be a flow instance. If the cut-condition holds and (G, Σ ) has no odd-K5 minor then there exists a fractional flow. There exists an algorithmic counterpart to this result, namely, Theorem 4. Let (G, Σ , w) be a flow instance. There exists an algorithm that will return, in polynomial time in the size of the flow instance, one of the following three outcomes: (a) a fractional flow y, (b) a cut δ(U ) that violates the cut-condition, (c) an odd-K5 minor. In the case of outcome (c) the algorithm tells us what sequence of minor operations are required to get odd-K5 from (G, Σ ). This is a special case of a result in [16]. Thus in polynomial time we either find a fractional flow, or find a reason why the flow does not exist, or find a certificate that indicates that we are outside the scope of Theorem 3.

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Fig. 4. Flow instance: demand edges are dashed, for every edge e, we = 1.

1.3. Integer flows in the Eulerian case Given a flow instance (G, Σ , w) we say that the weights are Eulerian if for every vertex v , w δ({v}) is an even number (recall that  all weights are non-negative integers). Observe that the example  in Fig. 2 is not Eulerian as for every vertex v we have w δ(v) = 3 but the example in Fig. 3 is Eulerian as in that case w δ(v) = 4 for every vertex v . A natural question is: Given a flow instance with Eulerian weights when do we have an integer flow? It was proved in [12] that as long as our flow instance does not ‘‘contain’’ the bad example in Fig. 3 there exists an integer flow, namely we have,





Theorem 5. Let (G, Σ , w) be a flow instance where w is Eulerian. If the cut-condition holds and (G, Σ ) has no odd-K5 minor then there exists an integer flow. Consider a flow instance (G, Σ , w) where the cut-condition holds and where (G, Σ ) has no odd-K5 minor. Clearly 2w is Eulerian and the cut-condition holds for (G, Σ , 2w). It follows from Theorem 5 that there exists an integer flow y for (G, Σ , 2w). Hence, 21 y is a flow for (G, Σ , w). Thus, Theorem 5 implies Theorem 3. However, we do not know of any algorithmic counterpart to Theorem 5. Open Problem 6. Let (G, Σ , w) be a flow instance where w is Eulerian. Find an algorithm that will return, in polynomial time in the size of the flow instance, one of the following three outcomes: (a) an integer flow y, (b) a cut δ(U ) that violates the cut-condition, (c) an odd-K5 minor. 1.4. From flow problems to minimax relations Consider a signed graph (G, Σ ). Recall that a circuit C is odd if |C ∩ Σ | is odd. We define τ (G, Σ ) to be the minimum number of edges needed to intersect every odd circuit of (G, Σ ). We define ν(G, Σ ) as the cardinality of the largest collection of pairwise (edge) disjoint odd circuits of (G, Σ ). Clearly, τ (G, Σ ) ≥ ν(G, Σ ). We say that (G, Σ ) packs if equality holds, i.e. τ (G, Σ ) = ν(G, Σ ). Consider the flow instance (G, Σ , w) in Fig. 4. There exists an integer flow y where yC1 = 1

for C1 = {yd, da, ay}

yC2 = 1

for C2 = {dz , zb, bd}

yC3 = 1

for C3 = {ac , cy, yz , zx, xb, ba}

and yC = 0 for all other circuits C ∈ C . Since |Σ | = 3, τ (G, Σ ) ≤ 3. Since all capacities/demands are 1 and since y is integer, C1 , C2 , C3 are pairwise edge disjoint. It follows that ν(G, Σ ) ≥ 3. Hence, (G, Σ ) packs. This works in general. Remark 7. Consider a flow instance (G, Σ , w) where we = 1 for every e. Then (G, Σ ) packs if there exists an integer flow. Indeed, suppose there exists an integer flow y. Then {C : yC = 1} is a collection of |Σ | circuits that are pairwise edge disjoint. Since Σ intersects all odd circuits, (G, Σ ) packs. There is an analogue of the previous remark for fractional flows. For a signed graph (G, Σ ) define,

 ν (G, Σ ) = max ∗

 

C odd circuit

yC :



yC ≤ 1 : ∀e ∈ E (G), y ≥ 0 .

e∈C odd circuit

It can be readily checked (using linear programming duality for instance) that τ (G, Σ ) ≥ ν ∗ (G, Σ ). We say that (G, Σ ) fractionally packs if equality holds, i.e. τ (G, Σ ) = ν ∗ (G, Σ ). Remark 8. Consider a flow instance (G, Σ , w) where we = 1 for every e. Then (G, Σ ) fractionally packs if there exists a fractional flow.

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5

Fig. 5. Fano matroid.

Consider a flow instance (G, Σ , w) where we = 1 for all e ∈ E (G). Suppose that Γ is a minimum cardinality set that intersects every odd circuit of (G, Σ ). Then it can be readily checked that Γ is a signature of (G, Σ ). Moreover, the cutcondition holds for (G, Γ , w) for otherwise, there exists a cut δ(U ) for which the signature Γ △δ(U ) has smaller cardinality than Γ , a contradiction. Suppose (G, Σ ) has no odd-K4 minor. Then (G, Γ ) has no odd-K4 minor and Theorem 1 implies that (G, Γ , w) has an integer flow. It follows from Remark 7 that (G, Γ ), respectively (G, Σ ), packs. Thus we have that, Theorem 9. A signed graph packs if it does not have an odd-K4 minor. Similarly, Remark 8 and Theorem 3 imply, Theorem 10. A signed graph fractional packs if it does not have an odd-K5 minor. Recall that a graph is Eulerian if every vertex has even degree. Similarly, Remark 7 and Theorem 5 imply, Theorem 11. A signed graph packs if it is Eulerian and it does not have an odd-K5 minor. It can be readily checked that Theorems 1, 3 and 5 are in fact equivalent to respectively Theorems 9, 10 and 11. 2. Flows in matroids In the previous section we stated sufficient conditions for the odd circuits of a signed graph to pack or fractionally pack. We wish to investigate when these results can be generalized from graphs to binary matroids. This will lead to two tantalizing conjectures by Paul Seymour. 2.1. A primer on matroids In the interest of making this survey accessible to people without background in matroid theory we present here the matroid related definitions we require. Consider a 0, 1 matrix A. The binary matroid M represented by A has elements E (M ) corresponding to the column indices of A. A subset C of elements is a cycle of M if C intersects each row with even parity.5 An inclusion-wise minimal cycle is a circuit. As an example consider the matrix,

A=

1 2 3 1 0 0 0 1 0 0 0 1



4 0 1 1

5 1 1 0

6 1 0 1

7  1 . 1 1

The matroid M represented by A has elements {1, 2, 3, 4, 5, 6, 7}. Its circuits correspond to the lines and the complements of the lines in Fig. 5. For instance {4, 5, 6} is a cycle as it intersects each row of A with even parity. Moreover, it is a circuit as no proper subset of {4, 5, 6} is a cycle. M is known as the Fano matroid and is denoted F7 . Let G be a graph and suppose that A is a matrix where the span of the rows of A is the cut space of G.6 For instance we can pick A to be the vertex-edge incidence matrix of G. Then the cycles of the matroid M represented by A will be the sets that intersect all cuts with even parity, i.e. they correspond to the cycles of G.7 We then say that M is the graphic matroid of G and denote it by cycle(G). Suppose that A is a matrix where the span of the rows of A is the cycle space of G. Then the cycles of the matroid M represented by A will be the sets that intersect all cycles with even parity, i.e. they correspond to the cuts of G. We then say that M is the co-graphic matroid of G and denote it by cut (G).

5 We identify a 0, 1 vector x with the set {i : x = 1}. i 6 We are considering the span of A over the two element field. Equivalently, the span of A is all the sets that can be obtain by symmetric difference of sets corresponding to rows of A. 7 A cycle of a graph G is a subset of edges that induces an Eulerian subgraph.

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Given a binary matroid M, a set of elements B is a cocycle if it intersects all cycles with even parity. For instance if M = cycle(G) the cocycles are the cut of G and if M = cut (G) then the cocycles are the cycles of G. The dual M ∗ of a binary matroid M is the matroid where the cycles of M ∗ are exactly the cocycles of M. For instance the dual of cycle(G) is cut (G) and the dual of cut (G) is cycle(G). Let M be a binary matroid and I , J be disjoint subsets of E (M ). Then M /I \J is the binary matroid with cycles of the form C − I where C ∩ J = ∅ and C is a cycle of M. We say that M /I \J is the minor of M obtained by contracting elements I and deleting elements J. Thus deleting J destroys the cycles of M using any element of J and contracting I removes elements in I from all cycles. Observe that cycle(G)/I \J is equal to cycle(G/I \J ), i.e. for graphic matroids the notion of graph and matroid minors coincide. 2.2. Packing odd cycles in signed matroids A signed matroid consists of a pair (M , Σ ) where M is a binary matroid and where Σ is a subset of the elements of M, i.e. Σ ⊆ E (M ). A set of elements B of M is odd if |B ∩ Σ | is odd and it is even otherwise. In particular, an element e is odd if e ∈ Σ . For instance for (F7 , E (F7 )), the odd circuits correspond to the three point lines in Fig. 5. We denote this signed matroid by L7 . In the context of signed matroids, an odd-K4 is the signed matroid obtained from the graphic matroid of K4 where all elements are odd, i.e. it is (cycle(K4 ), E (K4 )). Given a signed matroid (M , Σ ), we define τ (M , Σ ) as the minimum number of elements needed to intersect every odd circuit, and we define ν(M , Σ ) as the cardinality of the largest collection of pairwise disjoint odd circuits. We say that (M , Σ ) packs if τ (M , Σ ) = ν(M , Σ ). For instance, ν(L7 ) = 1 as any two lines in Fig. 5 intersect and τ (L7 ) = 3 as we need three points to intersect all these lines. In particular, L7 does not pack. Observe that if M = cycle(G), then (M , Σ ) packs if and only if (G, Σ ) packs. In particular, the signed matroid odd-K4 does not pack. We need to extend the notion of minors from signed graphs to signed matroids. Namely, we define the following operations on a signed matroid (M , Σ ):

• Deleting an element e: replacing (M , Σ ) by (M \e, Σ − {e}); • Contracting an element e ̸∈ Σ : replacing (M , Σ ) by (M /e, Σ ); • Resigning: replacing (M , Σ ) by (M , Σ △B) where B is a co-cycle of M. A signed matroid (N , Γ ) is a minor of (M , Σ ) if it can be obtained from (M , Σ ) by repeatedly applying the aforementioned operations. Theorem 9 says that a signed graph packs if it does not have an odd-K4 minor. Remarkably this theorem extends to signed matroids. Namely, Seymour [21] proved the following stunning result. Theorem 12. A signed matroid packs if it does not have an odd-K4 minor. This theorem is often stated in terms of binary clutters. A clutter H over a finite ground set E (H ) is a collection of distinct subsets of E (H ), where no subset is contained in another. We say that H is binary if for S1 , S2 , S3 ∈ H , S1 △S2 △S3 contains a set of H . Binary clutters and signed matroids are related by the following folklore remark, Remark 13. The following are equivalent for a clutter H , 1. H is a binary clutter, 2. H is the odd circuits of a signed matroid. We say that a clutter H packs if the minimum number of elements of the ground set needed to intersect all sets in H is equal to the maximum number of pairwise disjoint elements of H . Because of Remark 13 we can see that Theorem 12 yields sufficient conditions for a binary clutter to pack. In fact Theorem 12 exactly characterizes the binary clutters that have the Strong Max-Flow Min-Cut property. 2.3. Fractionally packing odd cycles in signed matroids Consider a signed matroid (M , Σ ). We define,

 ν (M , Σ ) = max ∗

  C odd circuit

yC :



yC ≤ 1 : ∀e ∈ E (M ), y ≥ 0 .

e∈C odd circuit

We say that (M , Σ ) fractionally packs if τ (M , Σ ) = ν ∗ (M , Σ ). In the context of signed matroids, an odd-K5 is the signed matroid obtained from the graphic matroid of K5 where all elements are odd, i.e. it is (cycle(K5 ), E (K5 )). Theorem 10 says that a signed graph fractionally packs if it does not have an odd-K5 minor. However, this theorem does not extend to signed matroids. Indeed, we have, τ (L7 ) = 3 but ν ∗ (L7 ) = 37

(we can assign 31 to each of the 7 lines). Hence, L7 does not fractionally pack. Consider the binary clutter H with ground set corresponding to the edges of K5 and sets corresponding to the complements of all the cuts of K5 . By Remark 13, the sets of H correspond to the odd circuits of a signed matroid that we denote by C5 . No two edges intersect all complements of cuts

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of K5 but every triangle does, thus τ (C5 ) = 3. We have ν ∗ (C5 ) =

10 4

)

–

(we can assign

7 1 4

to the

  5 2

sets E (G) − δ(U ) where

|U | = 2). Hence, C5 does not fractionally pack. Thus we have seen three examples of signed matroids that do not fractionally pack. The following conjecture of Seymour [23] proposes that every signed matroid that does not fractionally pack must contain one of these three examples, Flowing Conjecture. A signed matroid fractionally packs if it does not have any of the following minors: odd-K5 , L7 and C5 . Cornuéjols [6] has promised a $5000 award for a proof or a counterexample to this conjecture. Theorem 10 proves the case of the conjecture for the case of signed matroids (M , Σ ) where M is graphic. Before we investigate the case where M is co-graphic, i.e. where M = cut (G) for some graph G, we need a few definitions. A pair (G, T ) where T ⊆ V (G) and |T | is even is a graft and T is the set of terminals. A T -cut is a cut of the form δ(U ) where |U ∩ T | is odd. A T -join is a subset B of the edges of G that forms a subgraph where the vertices of odd degree correspond exactly to T . We leave the following remark as an easy exercise. Remark 14. Let (G, Σ ) be a signed graph and let T be the vertices of odd degrees of the subgraph formed by the edges in Σ . Then for every cut δ(U ), |δ(U ) ∩ Σ | is odd if and only if |U ∩ T | is odd. Consider a signed matroid (M , Σ ) where M is the co-graphic matroid of a graph G. It can be readily checked that none of odd-K5 , L7 and C5 are minors of (M , Σ ). Then the conjecture says that τ (M , Σ ) = ν ∗ (M , Σ ). Define T as in Remark 14. By that remark, the odd circuits of (M , Σ ) are T -cuts. It can be readily checked that a minimal set of edges that intersects every T -cut must be a T -join. Thus the conjecture would imply that there exists a T -join B where,

 |B| = max

 

δ(U ): T -cut

yU :



yU ≤ 1 : ∀e ∈ E (G), y ≥ 0 ,

e∈δ(U ): T -cut

i.e. the size of the smallest T -join is equal to the value of a fractional packing of T -cuts. This is in fact the case and follows from Edmond’s matching polytope Theorem [10]. Thus the Flowing Conjecture holds for both graphic and co-graphic signed matroids. In [7] we used Seymour’s decomposition of regular matroids [22] to prove the following common generalization, Theorem 15. The Flowing Conjecture holds for signed matroids (M , Σ ) where M does not contain the Fano matroid. Note, in that theorem we exclude the Fano as a minor of M not as a minor of the signed matroid (M , Σ ). 2.4. Packing odd cycles in Eulerian signed matroids Recall that a graph is Eulerian if every vertex has even degree. Equivalently, a graph is Eulerian if every cut contains an even number of edges. We say that a binary matroid is Eulerian if every co-cycle contains an even number of elements. Note that the matroid cycle(G) is Eulerian if and only if G is Eulerian. We will say that a signed matroid (M , Σ ) is Eulerian whenever, M is Eulerian. In particular, the signed matroid odd-K5 is Eulerian. Since the co-cycles of F7 are given by the complement of the lines in Fig. 5, F7 is Eulerian. Finally note that the co-cycles of C5 are given by the even cycles of K5 , hence, C5 is Eulerian as well. Theorem 11 says that an Eulerian signed graph packs if it does not have an odd-K5 minor. A generalization to signed matroids will be proposed in the Cycling Conjecture. Since odd-K5 , L7 , and C5 do not fractionally pack, they do not pack. Thus we have three examples of Eulerian signed matroids that do not pack. There is another important example however, but we require a few definitions before we can present it. The chromatic index of a graph G is the minimum number of colours we can assign to the edges of G so that no two edges of the same colour are adjacent to the same vertex. A postman set of a graph G is a subset B of the edges with the property that the subgraph formed by the edges B has the same vertices of odd degree as G. Consider the binary clutter with groundset corresponding to the edges of the Petersen graph and sets corresponding to postman sets of the Petersen graph. By Remark 13 the sets of H correspond to the odd circuits of a signed matroid that we denote by P10 . It can be readily checked that we need exactly three edges to intersect all Postman sets of the Petersen graph, hence, τ (P10 ) = 3 (pick three edges incident to the same vertex). However, ν(P10 ) < 3 for otherwise there would exist edge disjoint postman sets B1 , B2 , B3 . As all vertices have degree 3, B1 , B2 , B3 would have to be matchings. This would imply that the Petersen graph has chromatic index 3, a contradiction. Thus P10 does not pack. The cocycles of P10 correspond to cuts δ(U ) of the Petersen graph where |U | is even. Since all vertices have degree three, |δ(U )| is even. In particular, P10 is Eulerian. Thus we have seen four examples of Eulerian signed matroids that do not pack. The following conjecture of Seymour [23] (see also [20]) proposes that every Eulerian signed matroid that does not pack must contain one of these four examples. Cycling Conjecture. An Eulerian signed matroid packs if it does not have any of the following minors: odd-K5 , L7 , C5 and P10 . Theorem 11 proves the case of the conjecture for the case of signed matroids (M , Σ ) where M is graphic. Suppose that we consider an Eulerian signed matroid (M , Σ ) where M is the co-graphic matroid of a graph G. The Eulerian condition requires that every co-cycle of cut (G) be even, i.e. that every cycle of G is even, or equivalently that G be bipartite. It can be readily checked that none of odd-K5 , L7 , C5 and P10 are minors of (M , Σ ). Thus the conjecture says that (M , Σ ) should

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pack, i.e. the minimum number of edges needed to intersect every odd cut is equal to the maximum number of pairwise disjoint odd-cut. Because of Remark 14 this can be stated as the following theorem [24], Theorem 16. Let (G, T ) be a graft where G is bipartite. Then the maximum number of pairwise disjoint T -cuts is equal to the minimum size of a T -join. Thus, the Cycling Conjecture holds for both graphic and co-graphic signed matroids. However, the conjecture is likely very hard as it would imply some far reaching generalizations of the 4-colour Theorem. We will present several special cases of the Cycling Conjecture. Let (G, T ) be a graft and consider a partition of V (G) into subsets V1 , . . . , Vr such that for each i ∈ [r ] the induced subgraph G[Vi ] is connected and |Vi ∩ T | is odd. Let G′ be the simple graph with vertex set [r ] and where ij ∈ E (G′ ) if and only if there exists an edge of G with one end in Vi and one end in Vj . A subgraph of G′ is called a T -minor of G. Observe that if a graph is a T -minor of G then it is minor of G as well, however the converse is not true in general. The following is a special case of the Cycling Conjecture, Conjecture 17. Let (G, T ) be a graft where all T -cuts have the same parity. If G does not contain the Petersen graph as a T -minor, then the size of the minimum T -cut is equal to the maximum number of pairwise T -joins. Consider the special case where G is a bridgeless cubic graph that does not contain the Petersen graph and where T is the set of all vertices of G. Then all T -cuts have the same parity. Since G is bridgeless, the size of the minimum T -cuts is 3. It would follow from Conjecture 17 that there exist 3 disjoint T -joins. Thus this conjecture would imply the recently proved conjecture of Tutte [26,19,11] which states that cubic bridgeless graphs with no Petersen minor have chromatic index three. ⃗ a map φ : E (G ⃗ ) → Z is a nowhere zero k-flow if −k < φ(e) < k and φ(e) ̸= 0 for all e ∈ E (G ⃗) Given a directed graph G, and for every v ∈ V (G), φ(δ + (v)) = φ(δ − (v)).8 If a digraph has a nowhere zero k-flow φ then so does any digraph obtained by changing the direction of an edge e as it suffices to replace φ(e) by −φ(e) for that edge. A graph is said to have a nowhere zero k-flow if some (resp. every) orientation has a nowhere zero k-flow. It can be easily shown that a graph has a nowhere zero 4-flow if and only if it contains three pairwise disjoint postman sets. Thus Conjecture 17 would also imply, the following conjecture of Tutte [28], Conjecture 18. Every bridgeless graph not containing the Petersen graph as a minor has a nowhere zero 4-flow. Choosing a k-regular planar graph in Conjecture 17 we get the following specialization, Conjecture 19. Let G be a k-regular planar graph. Then G has chromatic index k if and only if for all U ⊆ V (G) where |U | is odd, we have |δ(U )| ≥ k. Observe that necessity in the previous proposition is obvious as each of the k colour classes forms a matching which implies that they each contain an edge in δ(U ) for all |U | odd. For k = 3 this conjecture says that planar bridgeless cubic graphs have chromatic index 3. By a result of Tait [25] this is equivalent to the 4-colour Theorem. The cases k = 4 and k = 5 were proved in [15]. The case k = 6 can be found in [9], the case k = 7 in [3] and the case k = 8 in [4]. We say that H is an odd minor of G if H can be obtained from G by deleting edges and then contracting all the edges on a cut. Another special case of the Cycling Conjecture is as follows, Conjecture 20. Consider a graph G that does not contain K5 as an odd minor. If the length of the shortest odd cycle is k, then there exist cuts B1 , . . . , Bk such that every edge e is in at least k − 1 of B1 , . . . , Bk . Consider the case of loopless planar graph for this latest conjecture. Then the length of the shortest odd cycle is at least 3. It follows that there exist cuts Bi = δ(Ui ) for i = 1, 2, 3 such that every edge e is contained in B1 ∪ B2 . Therefore, U1 ∩ U2 , U1 ∩ U 2 , U 1 ∩ U2 , U 1 ∩ U 2 are all stable sets. As these sets partition the vertex set it follows that G is 4-colourable. Thus Conjecture 20 also implies the 4-colour Theorem. 2.5. A new case of the cycling conjecture We have seen that the Cycling Conjecture holds for graphic and co-graphic signed matroids. However, proving the conjecture in its entirety seems difficult as it contains far reaching generalizations of the 4-colour Theorem. To find interesting generalizations of the graphic and co-graphic case we need to find generalizations of graphic and co-graphic matroids. Let (G, Σ ) be a signed graph and let A be a 0, 1 matrix where the row span of A is equal to the cut space of G. Recall that the matroid represented by A is cycle(G). Let A′ be obtained from A by adding an additional row R that corresponds to the characteristic vector of a set Σ ⊆ E (G). Let M be the matroid represented by A′ . Let C be a cycle of M. Then C has to intersect

8 δ + (v) is the set of edges with tail v and δ − (v) is the set of edges with head v .

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Fig. 6. (left) L7 as a signed graft: odd edges are dashed, terminals are square vertices, (right) lines of the Fano.

all cuts of G with even parity. In particular, C must be a cycle of G. Moreover, because of row R of A, C must intersect Σ with even parity. It follows that C is an even cycle of (G, Σ ). We denote this matroid by ecycle(G, Σ ) and say that it is the even cycle matroid of (G, Σ ). Note that if Σ = ∅ then ecycle(G, Σ ) = cycle(G), thus the class of even-cycle matroids generalizes the class of graphic matroids. Let M be a binary matroid and let e ∈ E (M ). A set of the form C − {e} where C is a circuit of M using e is an e-path of M. If M = cycle(G) and e is an edge with endpoints s, t, then the e-paths of M are exactly the st-paths of G. If M = cut (G) and e = st, then e-paths of M are exactly the st-cuts of G. Suppose that M = ecycle(G, Σ ) and let e = st ∈ Σ . Note, we allow the case where s = t, i.e. where e is an odd loop. If s ̸= t then the e-paths of ecycle(G, Σ ) correspond to odd st-paths. If s = t then the e-paths of ecycle(G, Σ ) correspond to odd cycles. Thus the e-paths of an even-cycle matroid correspond to the odd T -joins of a signed graph where |T | ≤ 2. We have the following extension of Remark 13, Remark 21. The following are equivalent for a clutter H , 1. H is a binary clutter, 2. H is the odd circuits of a signed matroid, 3. H is the e-paths of a binary matroid. It follows in particular, that the e-paths of an even-cycle matroid N correspond to the odd circuits of signed matroid (M , Σ ). We identify (M , Σ ) with the set of e-paths of N. Since for cycle(G) and e = st ∈ E (G), the e-path are st-paths, Menger’s Theorem implies that the e-path of cycle(G) packs. In particular, the Cycling Conjecture holds for e-paths of graphic matroids. Similarly, the Max Work Min Potential Theorem of Duffin [8] implies the Cycling conjecture for e-paths of co-graphic matroids. We recently proved the following [1,2]. Theorem 22. The Cycling Conjecture holds for the e-paths of even-cycle matroids. Let us restate this theorem. A signed graft is a triple (G, Σ , T ) where (G, Σ ) is a signed graph and (G, T ) is a graft. We say that a signed graft (G, Σ , T ) packs if the minimum number of edges needed to intersect all the odd T -joins of G is equal to the maximum number of pairwise disjoint odd T -joins of G. In Fig. 6 (left) we represent the odd circuits of L7 as the odd T -joins in a signed graft. For instance, {1, 4, 7} is an odd T -join and it corresponds to a line of the Fano. A signed graft is Eulerian if vertices v ̸∈ T have even degree and vertices v ∈ T all have the same parity as Σ . For instance, in Fig. 6, the vertices outside T have degree 4 which is even and vertices in T have degree three which is the same parity as the number of odd edges. The odd T -joins of a signed graft correspond to the odd circuits of a signed matroid. Thus we can translate the minor operations on a signed matroid for the corresponding signed graft. This leads to the following containment operation for a signed graft (G, Σ , T ):

• Deleting an element e: replacing (G, Σ , T ) by (G\e, Σ − {e}, T ); • Resigning: replacing (G, Σ , T ) by (G, Σ △δ(U ), T ) where δ(U ) is a cut and |U ∩ T | is even; • Contracting an element e ̸∈ Σ : replacing (G, Σ , T ) by (G/e, Σ , T ′ ) where T ′ = T − {u, v} if both or none of u, v are in T and T ′ = T − {u, v} ∪ {w} if exactly one of u, v is in T where w is the vertex obtained from e by contracting e. A signed graft (H , Γ , R) is a minor of (G, Σ , T ) if it can be obtained from (G, Σ , T ) by repeatedly applying the aforementioned operations. In the context of signed grafts, an odd-K5 is the signed graft (K5 , E (K5 ), ∅), i.e. an odd-K5 with no terminals. In that context L7 is the signed graft given in Fig. 6 (left). We can now restate Theorem 22, Theorem 23. If an Eulerian signed graft has at most two terminals and it does not contain odd-K5 or L7 as a minor then it packs. Observe that the Eulerian condition cannot be omitted. For instance the signed graft (K4 , E (K4 ), ∅) does not pack. However, none of the vertices are terminals and all the vertices have odd degree. Thus the Eulerian condition is clearly violated in this case. Similarly, consider the signed graft obtained from L7 in Fig. 6, by deleting the unique edge between the two terminals. It can be readily checked that it does not pack. However, the vertices in T have degree 2 but there are 3 odd edges. Thus the Eulerian condition is also violated in this case.

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2.6. Special cases of Theorem 23 We will show that we can derive many classical results as special cases of Theorem 23. Consider a signed graft (G, Σ , T ). When T = ∅, the condition that (G, Σ , T ) is Eulerian says that all vertices of G should have even degree, i.e. G should be Eulerian. Furthermore, in that case odd T -joins are odd cycles. Finally, the notion of minor for signed graphs and signed graft coincide as T = ∅. Hence, Theorem 23 for the case T = ∅ says that a signed graph packs if it is Eulerian and it does not have an odd-K5 minor, which was the statement of Theorem 11. In the next proposition [2] we indicate a number of classes of signed grafts that do not contain odd-K5 and do not contain L7 . In particular, for each of these classes, as long as the Eulerian condition holds, the minimum number of edges needed to intersect all odd T -joins is equal to the maximum number of pairwise disjoint odd T -joins. A blocking vertex (resp. blocking pair) of a signed graph is a vertex (resp. pair of vertices) that intersects every odd circuit. Proposition 24. Let (G, Σ , T ) be a signed graft where T = {s, t }. If any of (1)–(6) hold then (G, Σ , T ) does not contain as a minor odd-K5 or L7 , (1) (2) (3) (4) (5) (6)

There exists a blocking vertex, s, t is a blocking pair, Every inclusion-wise minimal odd T -join is connected, G is a plane graph with at most two odd faces, G is a plane graph with a blocking pair u, v where s, u, t , v appear on a facial cycle in this order, G has an embedding on the projective plane where every face is even and s, t are connected by an odd edge. As a corollary of Theorem 23 and Proposition 24(1) we obtain,

Corollary 25. Let (H , T ) be a graft with |T | ≤ 4. Suppose that every vertex of H not in T has even degree and that all the vertices in T have degrees of the same parity. Then the maximum number of pairwise disjoint T -joins is equal to the minimum size of a T -cut. In fact this result holds as long as |T | ≤ 8 [5]. As a corollary of Theorem 23 and Proposition 24(2) we obtain, Corollary 26 (Hu [17], Rothschild and Whinston [18]). Let H be a graph with vertices s1 , s2 , t1 , t2 where s1 ̸= t1 , s2 ̸= t2 , all of s1 , t1 , s2 , t2 have the same parity, and all the other vertices have even degree. Then the maximum number of pairwise disjoint paths that are between si and ti for some i = 1, 2, is equal to the minimum size of an edge subset whose deletion removes all s1 t1 and s2 t2 -paths. Consider G obtained as follows: (⋆) start from a plane graph with exactly two faces of odd length and distinct vertices s and t, and identify s and t. As a corollary of Theorem 23 and Proposition 24(3) we obtain, Corollary 27. Let G be a graph as in (⋆) and suppose that the length of the shortest odd circuit is k. Then there exist cuts B1 , . . . , Bk such that every edge e is in at least k − 1 of B1 , . . . , Bk . This proves Conjecture 20 for graphs of the form (⋆). 2.7. An open question Let (G, Σ ) be a signed graph and let A be a 0, 1 matrix where the row span of A is equal to the cycle space of G. Let A′ be obtained from A by adding an additional row R that corresponds to the characteristic vector of a set Σ ⊆ E (G). Let M be the matroid represented by A′ and let C be a cycle of M. Then C has to intersect all cycles of G with even parity. In particular, C must be a cut of G. Moreover, because of row R of A, C must intersect Σ with even parity. It follows that C is an even cut of (G, Σ ). We denote this matroid by ecut (G, Σ ) and say that it is the even cut matroid of (G, Σ ). Note that if Σ = ∅ then ecut (G, Σ ) = cut (G), thus the class of even-cut matroids generalizes the class of co-graphic matroids. Guenin [14] proved that the Flowing Conjecture holds for both the e-paths of even-cycle matroids and the e-paths of even-cut matroids. Moreover, Seymour [23] proved that if the Flowing Conjecture holds for the e-paths of a binary matroid M, then it holds for e-paths of its dual M ∗ . The following table indicates the known/open special cases for the Flowing and Cycling Conjecture. For each class of matroid, we indicate for what cases the conjecture is proved for the e-paths of a matroid in that class. In the following table 4CT refers to the 4-colour Theorem

Even cycle matroid Even cut matroid Dual of even cycle matroid Dual of even cut matroid

Flowing Conjecture

Cycling Conjecture

Proved Proved Proved Proved

Proved Open Open implies 4CT Open implies 4CT

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Proving the Cycling Conjecture for e-paths of duals of even-cycle or even-cut matroid seems hard as both cases would imply far reaching generalizations of the 4-colour Theorem. It is possible that the same technique used to prove Theorem 23 can be adapted to solve the following problem. Open Problem 28. Prove the Cycling Conjecture for the e-paths of even-cut matroids. In other words prove the last remaining open case that does not imply the 4-colour Theorem in the previous table. Acknowledgements The author’s research was supported by NSERC grant 238811 and ONR grant N00014-12-1-0049. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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