Nowhere-zero flows in signed graphs:A survey

Tomas Kaiser∗ Robert Lukot’ka† Edita Rollova ‡

Abstract

We survey known results related to nowhere-zero flows and related top-ics, such as circuit covers and the structure of circuits of signed graphs. Weinclude an overview of several different definitions of signed graph colour-ing.

Keywords: signed graph, bidirected graph, survey, integer flow, nowhere-zero flow, flow number, signed circuit, Bouchet’s Conjecture, zero-sum flow,signed colouring, signed homomorphism, signed chromatic number

1 Introduction

Nowhere-zero flows were defined by Tutte [62] as a dual problem to vertex-colouring of (unsigned) planar graphs. Both notions have been extended tosigned graphs. The definition of nowhere-zero flows on signed graphs naturallycomes from the study of embeddings of graphs in non-orientable surfaces, wherenowhere-zero flows emerge as the dual notion to local tensions. There is a closerelationship between nowhere-zero flows and circuit covers of graphs as everynowhere-zero flow on a graph G determines a covering of G by circuits. Thisrelationship is maintained for signed graphs, although a signed version of thedefinition of circuit is required.

∗Department of Mathematics, Institute for Theoretical Computer Science (CE-ITI), andEuropean Centre of Excellence NTIS (New Technologies for the Information Society), Universityof West Bohemia, Pilsen, Czech Republic. Supported by project GA14-19503S of the CzechScience Foundation. Email: kaisert@kma.zcu.cz.†Department od Computer Science, Faculty of Mathematics, Physics, and Informatics,

Comenius University, Bratislava, Slovakia. Supported by project VEGA 1/0876/16. Email:lukotka@dcs.fmph.uniba.sk.‡European Centre of Excellence NTIS (New Technologies for the Information Society), Uni-

versity of West Bohemia, Pilsen, Czech Republic. Partially supported by project GA14-19503Sof the Czech Science Foundation and by project LO1506 of the Czech Ministry of Education,Youth and Sports. Email: rollova@ntis.zcu.cz.

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The area of nowhere-zero flows on (signed) graphs recently received a lot ofattention thanks to the breakthrough result of Thomassen [60] on nowhere-zero 3-flows in unsigned graphs. The purpose of this paper is to capture the current stateof knowledge regarding nowhere-zero flows and circuit covers in signed graphs.To cover the latest developments in this active area, we decided to include somepapers that are currently still in the review process and only available from thearXiv. Clearly, until the papers are published, the results have to be taken withcaution.

An extensive source of links to material on signed graph is the dynamic sur-vey of Zaslavsky [78]. For the theory of unsigned graphs in general, see [9] or[23]. Nowhere-zero flows and circuit covers in unsigned graphs are the subject ofZhang [79]; see also [80].

Graphs in this survey may contain parallel edges and loops. A circuit is aconnected 2-regular graph.

A signed graph is a pair (G,Σ), where Σ is a subset of the edge set of G. Theedges in Σ are negative, the other ones are positive. An unbalanced circuit in(G,Σ) is an (unsigned) circuit in G that has an odd number of negative edges. Abalanced circuit in (G,Σ) is an (unsigned) circuit in G that is not unbalanced. Asubgraph of G is unbalanced if it contains an unbalanced circuit; otherwise, it isbalanced. A signed graph is all-negative (all-positive) if all its edges are negative(positive, respectively).

Signed graphs were introduced by Harary [29] as a model for social networks.Zaslavsky [76] then employed the following kind of equivalence on the class ofsigned graphs. Given a signed graph (G,Σ), switching at a vertex v is the inver-sion of the sign of each edge incident with v. Thus, the signature of the resultinggraph is the symmetric difference of Σ with the set of edges incident with v.Signed graphs are said to be equivalent if one can be obtained from the other bya series of switchings. It is easy to see that equivalent signed graphs have thesame sets of unbalanced circuits and the same sets of balanced circuits.

Given a partition {A,B} of the vertex set of a graph G, we let [A,B] denotethe set of all edges with one end in A and one in B. Harary [29] characterisedbalanced graphs:

Theorem 1. A signed graph (G,Σ) is balanced if and only if there is a setX ⊆ V (G) such that the set of negative edges is precisely [X, V (G)−X].

It follows from the characterisation that a signed graph is balanced if it isequivalent to an all-positive signed graph. We say that a graph is antibalanced ifit is equivalent to an all-negative signed graph.

The role of circuits in unsigned graphs is played by signed circuits (known tobe the circuits of the associated signed graphic matroid [75]). A signed circuitbelongs to one of two types (cf. Figure 1):

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(a) A balanced circuit. (b) A (short) barbell.

(c) A (long) barbell.

Figure 1: Signed circuits. Dashed lines indicate negative edges.

(a) A consistently ori-ented edge.

(b) An introvertededge.

(c) An extrovertededge.

Figure 2: Oriented edges of a signed graph.

• a balanced circuit,

• a barbell, i.e., the union of two unbalanced circuits C1, C2 and a (possiblytrivial) path P with endvertices v1 ∈ V (C1) and v2 ∈ V (C2), such thatC1 − v1 is disjoint from P ∪ C2 and C2 − v2 is disjoint from P ∪ C1.

If the path in a barbell is trivial, the barbell is sometimes called short ; oth-erwise it is long. If we need to speak about a circuit in the usual unsigned sense,we call it an ordinary circuit.

To obtain an orientation of a signed graph, each edge of (G,Σ) is viewed ascomposed of two half-edges. An orientation of an edge e is obtained by giving eachof the two half-edges h, h′ making up e a direction. We say that e is consistentlyoriented if exactly one of h, h′ is directed toward its endvertex. Otherwise, itis extroverted if both h and h′ point toward their endvertices, and introverted ifnone of them does. (See Figure 2.) We say that an oriented edge e is incomingat its endvertex v if its half-edge incident with v is directed toward v, and thate is outgoing at v otherwise.

An orientation (bidirection) of (G,Σ) is the assignment of an orientation toeach edge of G (in the above sense), in such a way that the positive edges areexactly the consistently oriented ones. An oriented signed graph is often called abidirected graph.

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2 Nowhere-zero flows

2.1 Introduction

Let Γ be an Abelian group. A Γ-flow in (G,Σ) consists of an orientation of (G,Σ)and a function ϕ : E(G)→ Γ such that the usual conservation law is satisfied —that is, for each vertex v, the sum of ϕ(e) over the incoming edges e at v equalsthe sum of ϕ(e) over the outgoing ones. A Z-flow is said to be a k-flow (wherek ≥ 2 is an integer) if |ϕ(e)| < k for each edge e. A Γ-flow is nowhere-zero if thevalue 0 is not used at any edge. If a signed graph (G,Σ) admits a nowhere-zeroZ-flow, then its flow number Φ(G,Σ) is defined as the least k such that (G,Σ)admits a nowhere-zero k-flow. Otherwise, Φ(G,Σ) is defined as ∞.

A signed graph is said to be flow-admissible if it admits at least one nowhere-zero Z-flow. Bouchet [10] observed the following characterisation in combinationwith [75].

Theorem 2. A signed graph (G,Σ) is flow-admissible if and only if every edgeof (G,Σ) belongs to a signed circuit of (G,Σ).

Thus, an all-positive signed graph is flow-admissible if and only if it is bridge-less. Nowhere-zero flows on signed graphs are, in fact, a generalization of thesame concept on unsigned graphs, because the definition of a flow for all-positivesigned graph corresponds to the usual definition of a flow on an unsigned graph.

This useful corollary directly follows from the previous theorem:

Corollary 3. A signed graph with one negative edge is not flow-admissible.

For 2-edge-connected unbalanced signed graphs the converse is also true [42].

Theorem 4. An unbalanced 2-edge-connected signed graph is flow-admissible ex-cept in the case when it contains an edge whose removal leaves a balanced graph.

Tutte [61] proved that an unsigned graph G admits a nowhere-zero k-flow ifand only if it admits a nowhere-zero Zk-flow. He proved, in fact, that the numberof nowhere-zero Γ-flows only depends on |Γ| and not on the structure of Γ. This isnot true for signed graphs in general. For instance, an unbalanced circuit admitsa nowhere-zero Z2-flow, but it does not admit any integer flow. It was, however,proved for signed graphs that if |Γ| is odd, then the number of nowhere-zeroΓ-flows does not depend on the structure of Γ [7]. In the same paper, Beck andZaslavsky further proved that if |Γ| is odd, the number of nowhere-zero Γ-flowson a signed graph (G,Σ) is a polynomial in |Γ|, independent of the actual group,extending the result of Tutte [62] for unsigned graphs (where arbitrary |Γ| isallowed).

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2.2 Bouchet’s conjecture

Nowhere-zero flows in signed graphs were introduced by Edmonds and John-son [24] for expressing algorithms on matchings, but systematically studied atfirst by Bouchet in the paper [10]. Bouchet also stated a conjecture that parallelsTutte’s 5-Flow Conjecture and occupies a similarly central place in the area ofsigned graphs:

Conjecture 5 (Bouchet). Every flow-admissible signed graph admits a nowhere-zero 6-flow.

The value 6 would be best possible since Bouchet showed that the Petersengraph with a signature corresponding to its triangular embedding in the projectiveplane (see Figure 3(a)) admits no nowhere-zero 5-flow. Another signed graphwith this property given in Figure 3(b) was found using a computer search byMacajova [34]; an infinite family of such graphs containing the one in Figure 3(c)was constructed by Schubert and Steffen [54]. (The other graphs in the family areobtained by arranging k unbalanced 2-circuits, k ≥ 5 odd, in a circular manneras in Figure 3(c).) Interestingly, no other examples of signed graphs with flownumber 6 appear to be known (except for signed graphs constructed from theknown ones by simple flow reductions).

In [10], Bouchet proved that the conjecture holds with the number 6 replacedby 216. The constant was improved by Zyka [82] to 30; according to [21], thesame result was obtained by Fouquet (unpublished). Currently, the best generalbound is due to DeVos [21]:

Theorem 6 (DeVos). Every flow-admissible signed graph admits a nowhere-zero12-flow.

2.3 Higher edge-connectivity

Assumptions about the connectivity of a graph usually allow for better boundson its flow number, or at least make the bounds easier to establish. In theunsigned case, for example, it is well-known that any 4-edge-connected graphadmits a nowhere-zero 4-flow, so in particular these graphs are not interestingwith respect to Tutte’s 5-Flow Conjecture.

For signed graphs, connectivity alone is not sufficient, as can be seen byCorollary 3: no signed graph with a single negative edge is flow-admissible. Thus,an assumption of flow-admissibility needs to be included.

Very recently, the following was shown by Cheng et al. [15] for 2-edge-connectedsigned graphs:

Theorem 7 (Cheng, Lu, Luo, Zhang). Every flow-admissible 2-edge-connectedsigned graph has a nowhere-zero 11-flow.

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(a) (b)

(c)

Figure 3: Signed graphs without nowhere-zero 5-flows. The example (c) givesrise to an infinite family.

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The flow number of 3-edge-connected flow-admissible signed graphs was boundedby 25 in [68], which was improved to 15 in [67], and to 9 in [71]. A further im-provement, based on the ideas of the proof of the Weak 3-Flow Conjecture [33,60] was obtained by Wu et al. [69]:

Theorem 8 (Wu, Ye, Zang and Zhang). Every flow-admissible 3-edge-connectedsigned graph admits a nowhere-zero 8-flow.

Khelladi [32] proved already in 1987 that any flow-admissible 4-edge-connectedsigned graph has a nowhere-zero 18-flow, and established Bouchet’s Conjecturefor flow-admissible 3-edge-connected graphs containing no long barbell. Bouchet’sConjecture was also established for 6-edge-connected signed graphs [70]. The bestpossible result for 4-edge-connected signed graphs was given by Raspaud andZhu [51] using signed circular flows (for further results regarding signed circularflows, see Section 2.6):

Theorem 9 (Raspaud and Zhu). Every flow-admissible 4-edge-connected signedgraph admits a nowhere-zero 4-flow.

2.4 Signed regular graphs

Most of the research in the area of signed regular graphs is focused on signedcubic graphs. Macajova and Skoviera [43] characterised signed cubic graphs withflow number 3 or 4.

Let (G,Σ) be a signed graph and let {X1, X2} be a partition of V (G). Ifset of positive edges of G is exactly [X1, X2], then the partition {X1, X2} isantibalanced. A signed graph (G,Σ) has an antibalanced partition if and only if itis antibalanced [29]. For connected antibalanced signed graphs, the antibalancedpartition is uniquely determined. The circuit (Z,Σ) with antibalanced partition{X1, X2} is half-odd if for some i = 1, 2, each component of Z − Xi is either Zitself or a path of odd length.

Theorem 10. Let G,Σ be a signed cubic graph. Then the following hold:

(i) (G,Σ) admits a nowhere-zero 3-flow if and only if it is antibalanced andhas a perfect matching,

(ii) (G,Σ) admits a nowhere-zero 4-flow if and only if an equivalent signedgraph contains an antibalanced 2-factor with all components half-odd whosecomplement is an all-negative perfect matching.

Group-valued nowhere-zero flows in signed cubic graphs were also investigatedin [43], with the following results:

Theorem 11. Let (G,Σ) be a signed cubic graph. Then the following holds:

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(i) (G,Σ) has a nowhere-zero Z3-flow if and only if it is antibalanced,

(ii) (G,Σ) has a nowhere-zero Z4-flow if and only if it has an antibalanced 2-factor,

(iii) (G,Σ) has a nowhere-zero Z2 × Z2-flow if and only if it G is 3-edge-colourable.

The integer flow spectrum S(G) of a graph G is the set of all t such that thereis some signature Σ of G such that Φ(G,Σ) = t. The analogous notion whereΦ is replaced by Φc (circular flow number, for the definition see Section 2.6) iscalled the flow spectrum and denoted by S(G).

Schubert and Steffen [54] investigated both kinds of flow spectra in signedcubic graphs. They proved the following (K3

2 denotes the graph on two verticeswith three parallel edges and no loops):

Theorem 12. If G is a cubic graph different from K32 , then we have the following

equivalences:

G has a 1-factor ⇐⇒ 3 ∈ S(G) ⇐⇒ 3 ∈ S(G) ⇐⇒ 4 ∈ S(G).

Moreover, if G has a 1-factor, then {3, 4} ⊆ S(G) ∩ S(G).

Moreover, there exists a graph with S(G) = {3, 4}: the cubic graph on 4vertices that has two double edges; and an infinite family of graphs with S(G) ={3, 4, 6} [54].

For general (2t+ 1)-regular graphs the following was proved in [54]:

Theorem 13. Let (G,Σ) be a signed graph. If G is (2t + 1)-regular and has a1-factor, then 3 ∈ S(G).

A rather different version of a ‘flow spectrum’ was introduced by Macajova andSkoviera [43]. Let the modular flow number Φmod(G,Σ) be the least integer k suchthat (G,Σ) admits a nowhere-zero Zk-flow. It is easy to see that Φmod(G,Σ) ≤Φ(G,Σ). The modular flow spectrum Smod(G,Σ) is the set of all k such that(G,Σ) admits a nowhere-zero Zk-flow.

It is shown in [43] that the modular flow spectrum of a signed cubic graphmay contain gaps. Namely, a signed cubic graph is given whose modular flowspectrum equals {3, 5, 6, 7, . . .}. On the other hand, it is also shown that bridgesare essential in this example:

Theorem 14. Let (G,Σ) be a bridgeless cubic signed graph with Φmod(G,Σ) = 3.Then (G,Σ) admits a nowhere-zero Γ-flow for any Abelian group Γ of order atleast 3, except possibly for Z2 × Z2. In particular, Smod(G,Σ) = {k : k ≥ 3}.

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2.5 Other classes of signed graphs

More detailed information about nowhere-zero flows has been obtained for someother classes of signed graphs, namely signed Eulerian, complete and completebipartite graphs, series-parallel graphs, Kotzig graphs, and signed graphs withtwo negative edges.

Unsigned Eulerian graphs enjoy the minimum possible flow number of 2. Thisis not the case with signed Eulerian graphs, for which the following was provedin [70]:

Theorem 15. A connected signed graph has a nowhere-zero 2-flow if and only ifit is Eulerian and the number of its negative edges is even.

Flows in signed Eulerian graphs were further investigated by Macajova andSkoviera [42] (cf. also the extended abstract [40]). Their results are summarisedby Theorem 16 below. Before we state it, we introduce one more notion. A signedEulerian graph is triply odd if it has an edge-decomposition into three Euleriansubgraphs sharing a vertex, each of which contains an odd number of negativeedges.

Theorem 16. Let (G,Σ) be a connected signed Eulerian graph. Then

(i) if (G,Σ) is flow-admissible, then it has a nowhere-zero 4-flow,

(ii) Φ(G,Σ) = 3 if and only if (G,Σ) is triply odd.

Observe that together with Theorem 15 and Corollary 3, this result implies afull characterisation of signed Eulerian graphs of each of the possible flow numbers(2, 3, 4 and ∞).

Another result of [42] is a characterisation, for any Abelian group Γ, of signedEulerian graphs admitting a nowhere-zero Γ-flow. Namely:

Theorem 17. Let (G,Σ) be a signed eulerian graph and let Γ be a nontrivialAbelian group. The following statements hold true. (a) If Γ contains an involu-tion, then G admits a nowhere-zero Γ-flow. (b) If Γ ∼= Z3, then (G,Σ) admitsa nowhere-zero Γ-flow if and only if G is triply odd. (c) Otherwise, G has anowhere-zero Γ-flow if and only if G is flow-admissible.

Macajova and Rollova [38] (cf. also the extended abstract [37]) determinedthe complete characterisation of flow numbers of signed complete and signedcomplete bipartite graphs:

Theorem 18. Let (Kn,Σ) be a flow-admissible signed complete graph. Then

(i) Φ(Kn,Σ) = 2 if and only if n is odd and |Σ| is even,

(ii) Φ(Kn,Σ) = 4 if and only if (Kn,Σ) is equivalent to (K4, ∅) or the signedcomplete graph K5 whose negative edges form a 5-cycle,

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(iii) Φ(Kn,Σ) = 3 otherwise.

The result of [38] concerning signed complete bipartite graphs involves a spe-cial family of graphs. The family R contains all signed bipartite graphs (K3,n,Σ)such that n ≥ 3 and each vertex is incident with at most one negative edge,except possibly for one vertex v of degree n, which is incident with at least oneand at most n − 2 negative edges; moreover, the two vertices of degree n otherthan v are incident with 0 and 1 negative edges, respectively.

Theorem 19. Let (Km,n,Σ) be a flow-admissible signed complete bipartite graph.Then

(i) Φ(Km,n,Σ) = 2 if and only if m,n, |Σ| are even,

(ii) Φ(Km,n,Σ) = 4 if and only if either (Km,n,Σ) ∈ R, or m = n = 4 and |Σ|is odd,

(iii) Φ(Km,n,Σ) = 3 otherwise.

Kaiser and Rollova [31] studied nowhere-zero flows in signed series-parallelgraphs and showed that Bouchet’s Conjecture holds in this class:

Theorem 20. Every flow-admissible signed series-parallel graph admits a nowhere-zero 6-flow.

They pointed out that although every unsigned series-parallel graph admitsa nowhere-zero 3-flow and so these graphs are uninteresting from the point ofview of the 5-Flow Conjecture, there are signed series-parallel graphs with flownumber 6 (such as the one in Figure 3(c) and the other graphs in this familyfound by Schubert and Steffen in [54]).

Another class of graphs for which Bouchet’s Conjecture has been verified(see [54]) is that of Kotzig graphs, namely 3-edge-colourable cubic graphs G withthe property that the removal of the edges of any one colour produces a Hamil-tonian circuit of G.

A different approach to Bouchet’s Conjecture was given by Rollova et al.in [52]. Consider a flow-admissible signed graph (G,Σ) with a minimum signature(that is a representative of the class of signed graphs equivalent to (G,Σ) withminimum number of negative edges). If |Σ| = 0, then the signed graph is all-positive and by Seymour’s 6-flow theorem [56], it admits a nowhere-zero 6-flow.By Corollary 3, |Σ| 6= 1. Thus, the least number of negative edges for whichBouchet’s Conjecture is open is 2. In [52] it was proved that the flow numberof (G,Σ) with |Σ| = 2 is at most 7, and if Tutte’s 5-flow conjecture holds, thenit is 6. Moreover, Bouchet’s Conjecture is true for cubic (G,Σ) with |Σ| = 2 ifG contains a bridge or if G is 3-edge-colourable or if G is a critical snark (thatis a graph that does not admits a nowhere-zero 4-flow such that G − e admitsa nowhere-zero 4-flow for any edge e of G). Furthermore, if G is bipartite, then(G,Σ) admits a nowhere-zero 4-flow.

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Figure 4: One member of a family of signed graphs for which the difference Φ−Φc

tends to 2.

2.6 Circular flows

Raspaud and Zhu [51] generalised the concept of unsigned circular flow to signedgraphs. For t ≥ 2 real, a circular t-flow in (G,Σ) is an R-flow (D,ϕ) such thatfor each edge e,

1 ≤ |ϕ(e)| ≤ t− 1.

If there is t such that (G,Σ) has a circular t-flow, then its circular flow numberΦc(G,Σ) is the infimum of all such t. The circular flow number is infinite if (G,Σ)admits no circular t-flow for any t.

It is natural to ask whether the relation between Φ and Φc for signed graphsis as close as in the unsigned case, where it is known that Φ(G) = dΦc(G)e forany unsigned graph G — in other words, the difference between Φ and Φc is lessthan 1. Raspaud and Zhu [51] established the following upper bound on Φ:

Theorem 21 (Raspaud and Zhu). For any flow-admissible signed graph (G,Σ),

Φ(G,Σ) ≤ 2dΦc(G,Σ)e − 1.

Furthermore, it was conjectured in [51] that Φ(G,Σ) − Φc(G,Σ) < 1 just asin the unsigned case. However, the conjecture was disproved by Schubert andSteffen [54] who constructed signed graphs (G,Σ) with the difference Φ(G,Σ)−Φc(G,Σ) arbitrarily close to 2. (See Figure 4 for one element of a family men-tioned in [54].) This result was strenthened by Macajova and Steffen [39] whoproved that the difference Φ(G,Σ)−Φc(G,Σ) can be arbitrarily close to 3. Fur-thermore, the result of [39] shows that Theorem 21 cannot be improved in general.

Bounds for the circular flow number of sufficiently edge-connected signedgraphs have been obtained by Raspaud and Zhu [51]:

Theorem 22 (Raspaud and Zhu). Let (G,Σ) be a flow-admissible signed graph.If G is 6-edge-connected, then Φc(G,Σ) < 4.

Thomassen [60] showed that with increasing edge-connectivity of a (non-signed) graph, the circular flow number approaches 2. Zhu [81] studied the exis-tence of circular t-flows in signed graphs for t close to 2. It turns out that that

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it is not the case with signed graphs. It is easy to see [81] that a graph that hasexactly 2k + 1 negative edges has circuilar flow number at least 2 + 1/k. There-fore it is natural to introduce the notion of essentially (2k+1)-imbalanced signedgraph, defined as a signed graph such that every equivalent signed graph haseither at least 2k + 1 negative edges, or an even number of negative edges. Theanalogue to the result of Thomassen [60], shows that if both edge-connecticvityand essential imbalancednes rises, then the circular flow number goes to 2 [81].

Theorem 23 (Zhu). Let k be a positive integer. If a signed graph is essentially(2k + 1)-imbalanced and (12k − 1)-edge-connected, then its circular flow numberis at most 2 + 1

k.

Another interesting aspect of circular flows is, that there exist gaps amongthe values of circular flow number of regular graphs [59] that separate bipartitegraphs from non-bipartite graphs. The gaps remain even in signed case [54]:

Theorem 24. Let t ≥ 1 be an integer and (G,Σ) be a signed (2t + 1)-regulargraph. If Φc(G,Σ) = r, then r = 2 + 1

tor r ≥ 2 + 2

2t−1 .

2.7 Zero-sum flows

Let G be a graph. A zero-sum flow in G (also called ‘unoriented flow’) is afunction f : E(G) → R− {0} such that for each vertex v, the values f(e) of alledges e incident with v sum to zero. For a positive integer k, a zero-sum flow fis a zero-sum k-flow if for each edge e, f(e) is an integer and 1 ≤ |f(e)| ≤ k− 1.The zero-sum flow number of a graph G, which will be denoted by Φz(G), isdefined similarly to the flow number, namely as the least k such that G admits azero-sum k-flow (possibly ∞).

Zero-sum flows are tightly connected with nowhere-zero flows on signed graphs.Indeed, a zero-sum k-flow in G is nothing but a nowhere-zero k-flow in the all-negative graph (G,E(G)). Conversely, we can find a nowhere-zero flow on (G,Σ)by finding a zero-sum flow of graph G′ created from (G,Σ) by subdividing allpositive edges with one vertex of degree 2 [3]. The 6-flow Conjecture, therefore,naturally translates to the following conjecture [3]:

Conjecture 25 (Zero-Sum Conjecture). If a graph G admits a zero-sum flow,then it admits a zero-sum 6-flow.

Much of the work regarding zero-sum flows pertains to regular graphs. Whilenot all 2-regular graphs admit a zero-sum flow, Akbari et al. [3] proved that anyr-regular graph with r ≥ 4 even has a zero-sum 3-flow, and any cubic graph hasa zero-sum 5-flow. This led to the following conjecture, stated in [1]:

Conjecture 26. Any r-regular graph with r ≥ 3 has a zero-sum 5-flow.

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It was shown in [1] that Conjecture 26 holds for odd r divisible by 3, andthat Φz(G) ≤ 7 for general r. This was improved in [4] where the conjecture wasproved for all r except r = 5. Furthermore, a mention in [53] indicates that thecase of 5-regular graphs has also been settled in the affirmative, namely in [72].If so, then Conjecture 26 is proved.

Wang and Hu [64] gave exact values of Φz(G) for certain r-regular graphs G:

• Φz(G) = 2 if either r ≡ 0 (mod 4), or r ≡ 2 (mod 4) and |E(G)| is even,

• Φz(G) = 3 if one of the following holds:

– r ≡ 2 (mod 4) and |E(G)| is odd,

– r ≥ 3 is odd and G has a perfect matching,

– r ≥ 7 is odd and G is bridgeless.

The zero-sum flow number is determined in [64] for the Cartesian product of anr-regular graph G with an s-vertex path Ps (r, s ≥ 2); namely, Φz(G× Ps) = 2 ifr is odd and s = 2, and Φz(G× Ps) = 3 in the other cases.

Further results concerning the zero-sum flow number of (not necessarily reg-ular) graphs include:

• Φz(G) ≤ 6 if G is bipartite bridgeless [3],

• Φz(G) ≤ 12 if G is hamiltonian [2].

As for particular graph classes, exact values of the zero-sum flow number areknown for graphs of the following types:

• wheels and fans [65],

• all induced subgraphs of the hexagonal grid [66],

• certain subgraphs of the triangular grid [65].

Results on the computational complexity of deciding the existence of a zero-sum flow were obtained in [20]. For positive integers k ≤ `, let us define a (k, `)-graph to be an unsigned graph with minimum degree at least k and maximumdegree at most `. It is known [3] that Conjecture 25 is equivalent to its restrictionto (2, 3)-graphs (that is, subdivisions of cubic graphs).

Let us summarise the findings of [20]:

Theorem 27. (i) There is a polynomial-time algorithm to determine whethera given (2, 4)-graph with O(log n) vertices of degree 4 has a zero-sum 3-flow,

(ii) it is NP-complete to decide whether a given (3, 4)-graph admits a zero-sum3-flow,

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(iii) it is NP-complete to decide whether a given (2, 3)-graph admits a zero-sum4-flow.

In [53], zero-sum flows are viewed in the more general setting of matrices. Azero-sum k-flow of a real matrix M is an integer-valued vector in its nullspacewith nonzero entries of absolute value less than k. From this point of view, azero-sum flow in a graph is a zero-sum flow of its vertex-edge incidence matrix.The main result of [53] proves the existence of a zero-sum k-flow of the incidencematrix for linear subspaces of dimension 1 and m, respectively, of the vector spaceFnq (where q is a prime power, 0 ≤ m ≤ n and k is a suitable integer depending

on q,m and n).

2.8 Structural results

Zaslavsky [76] proved that any signed graph (G,Σ) is associated with a matroid,called the frame matroid of (G,Σ) and denoted by M(G,Σ). (For backgroundon matroid theory, see, e.g., [49].) One way to define this matroid is to specifyits circuits, and these are precisely the signed circuits introduced in Section 1.Properties of M(G,Σ) are studied in [76] and subsequently in a number of papers.See, for instance, [57, 58] (decomposition theorems), [27, 50] (minors) and thereferences therein.

Let us call a Z-flow in a (signed or unsigned) graph nonnegative if all its valuesare nonnegative. A nonnegative Z-flow is irreducible if it cannot be expressed asa sum of two nonnegative Z-flows. Tutte [63] showed that in an unsigned graph,irreducible nonnegative Z-flows are precisely the circuit flows (elementary flowsalong circuits). Equivalently, any nonnegative Z-flow is a sum of circuit flows.

For signed graphs, a characterisation of irreducible flows appears in the (ap-parently unpublished) manuscript [14] and in [13]. Let us say that a signed graph(H,Θ) is essential if it satisfies the following properties:

• the degree of any vertex of H is 2 or 3,

• each vertex is contained in at most one circuit of H,

• for any circuit C of H, |E(C) ∩Θ| has the same parity as the number ofbridges of H, incident with C.

(See Figure 5 for an example.)Let C be a circuit in H and let Θ′ = Θ∩E(C). An orientation D of (C,Θ′) is

consistent at a vertex v of C if v is incident with precisely one incoming half-edge.Otherwise, D is inconsistent at v. A vertex is a source (sink) in an orientationof a signed graph if all incident half-edges are outgoing (incoming, respectively).

Given an essential signed graph (H,Θ), an essential flow in (H,Θ) is obtainedas follows:

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(a)

1

1

1

1

1

1

11

1

1

1

1

1

1

1

1

1

1

2

2

2 2 2

2

2

(b)

Figure 5: (a) An essential signed graph (H,Θ). (b) An essential flow in (H,Θ).

• choose an orientation of (H,Θ) with no sources nor sinks, such that theinduced orientation of any circuit C of H is inconsistent at any vertexincident with a bridge of H (we call this an essential orientation of (H,Θ)),

• assign flow value 2 to each bridge of H and value 1 to the other edges.

The main result of [14] reads as follows:

Theorem 28. Irreducible flows in a signed graph (G,Σ) are exactly the essentialflows on essential subgraphs of (G,Σ).

Independently, Macajova and Skoviera [41] proved the closely related resultthat nonnegative flows in signed graphs can be obtained as sums of elementaryflows along signed circuits at the cost of allowing half-integral values. Let Cbe a signed circuit in (G,Σ). An orientation of C will be called proper if itis an essential orientation of the essential subgraph C as defined above. The

15

characteristic flow of C is the function χC : E(G)→ Q with values

χC(e) =

1 if C is balanced and e ∈ E(C),12

if C is unbalanced and e is contained in an ordinary circuit of C,

1 if C is unbalanced and e is contained in the connecting path of C,

0 otherwise.

Theorem 29. Let (D,ϕ) be a Z-flow on a signed graph (G,Σ) such that all valuesof ϕ are nonnegative. Then D contains proper orientations of signed circuitsC1, . . . , Cm and positive integers αi (i = 1, . . . ,m) such that

ϕ =m∑i=1

αiχCi.

Chen and Wang [12] defined cuts in signed graphs and introduced the circuitand bond lattices of signed graphs.

3 Circuit covers

The Shortest Cycle Cover Conjecture of Alon and Tarsi [5] asserts that the edgeset of any (unsigned) bridgeless graph with m edges can be covered with circuitsof total length at most 7m

5. For background on the conjecture and its relation

to other problems see, e.g., [80, Chapter 14]. The best general upper bound tothe Shortest Cycle Cover Conjecture is due to Bermond et al. [8], and Alon andTarsi [5], who independently proved that any bridgeless graph admits a circuitcover of total length at most 5m

3.

The signed version of the problem was introduced by Macajova et al. [35]. Ina signed graph (G,Σ), it is natural to seek a cover of the entire edge set by signedcircuits (a signed circuit cover) of small length. Here, the length of a signedcircuit is the number of its edges, and the length of the cover is defined as thesum of lengths of its elements. Similarly as in the case of flows, for a signed circuitcover of G to exist, each edge has to be contained in a signed circuit, which isequivalent to (G,Σ) being flow-admissible (see Theorem 2). The authors of [35]obtained the following bounds:

Theorem 30. Let (G,Σ) be a flow-admissible signed graph with m edges. Thenthere is a signed circuit cover of (G,Σ) of length at most 11m. Moreover, if G isbridgeless, then it admits a signed circuit cover of length 9m.

An application of Theorem 29 given in [41] yields the following result:

Theorem 31. Let (G,Σ) be a signed graph with m edges. If (G,Σ) admits anowhere-zero k-flow, then it has a signed circuit cover of length at most 2(k−1)m.

16

Currently the best bound for this problem is due to Cheng et al. [16]. Let thenegativeness of a signed graph be the minimum number of negative edges in anyequivalent signed graph.

Theorem 32. Let (G,Σ) be a flow-admissible signed graph with n vertices, medges and negativeness ε > 0. Then the following holds:

(i) (G,Σ) has a signed circuit cover of length at most

m+ 3n+ min

{2m

3+

4ε

3− 7, n+ 2ε− 8

}≤ 14m

3− 5ε

3− 4.

(ii) If G is bridgeless and ε is even, then (G,Σ) admits a signed circuit coverof length at most

m+ 2n+ min

{2m

3+ε

3− 4, n+ ε− 5

}.

4 Relation to colouring

As indicated in Section 1, nowhere-zero flows on signed graphs are dual to localtensions on non-orientable surfaces (see [22]). For an embedded directed graphG and a function φ : E(G) → A, we say that φ is a local tension if the heightof every facial walk is 0 (the height of a facial walk for a given orientation ofthe surface, is the sum of values on edges of the face directed consistently withthe orientation minus the sum of values on the edges directed inconsistently).Since the surface is non-orientable, the dual graph G∗ of a directed graph G willbe bidirected, and the condition on the facial walk of a local tension of G willtranslate to the conservation law at a vertex of G∗. This duality can be used toshow that the signed Petersen graph of Figure 3(a) does not admit any nowhere-zero 5-flow, because its dual in the projective plane is K6, which does not admitany 5-local-tension.

The flow-colouring duality does not hold for signed graphs for any of theknown definitions of signed colourings. However, given the importance of theduality in the unsigned case, it will perhaps be useful to include an overview ofthe essential definitions of signed colourings and the appropriate literature forcompleteness. The first definition is due to Zaslavsky [75]:

Signed colouring according to Zaslavsky. If (G,Σ) is a signed graph, a(signed) colouring of (G,Σ) in k colors, or in 2k + 1 signed colors is a mapping

f : V (G)→ {−k,−k + 1, . . . ,−1, 0, 1, . . . , k − 1, k}.

A colouring is zero-free if it never uses the value 0. The (signed) colouringis proper if for every edge uv f(u) 6= f(v) · Σ(uv). Zaslavsky further defined

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the chromatic polynomial ξG(2k + 1) of a signed graph G to be the functioncounting the number of proper signed colourings of G in k colours. The balancedchromatic polynomial ξbG(2k) is the function which counts the zero-free propersigned colourings in k colours. Then the chromatic number γ(G) of G, accordingto Zaslavsky, is the smallest nonnegative integer k for which ξG(2k + 1) > 0.Furthermore, the strict chromatic number γ∗(G) of G is the smallest nonnegativeinteger k such that ξbG(2k) > 0. Neither of Zaslavsky’s definitions of chromaticnumber is a direct extension of the usual chromatic number of an unsigned graph.These signed colourings were further investigated by Zaslavsky in the series ofpapers [73, 74, 77], and recently by Beck et al. [6], by Davis [19], and by Schweseret al. [55].

Signed colouring according to Macajova, Raspaud and Skoviera. Amodification of Zaslavsky’s definition of a signed colouring rised to a ’similar’definition of signed colouring introduced by Macajova, Raspaud,Skoviera in [36].Let, for each n ≥ 1, Mn ⊆ Z be a set Mn = {±1,±2, . . . ,±k} if n = 2k, andMn = {0,±1,±2, . . . ,±k} if n = 2k + 1. A proper colouring of a signed graphG that uses colours from Mn is an n-colouring. Thus, an n-colouring of a signedgraph uses at most n distinct colours. Note that if G admits an n-colouring, thenit also admits an m-colouring for each m ≥ n. The smallest n such that G admitsan n-colouring is the signed chromatic number of G and is denoted by ξ(G). It iseasy to see that the chromatic number of a balanced signed graph coincides withthe chromatic number of its underlying unsigned graph, and hence this definitionof chromatic number differs from Zaslavsky’s. Moreover, ξ(G) = γ(G) + γ∗(G).Despite the fact that this definition is very recent, it has already been studied byJin et al. [30] and by Fleiner et al. [25].

Signed colourings via signed homomorphisms according to Naserasr,Rollova and Sopena. A chromatic number for a graph can be defined usinggraph homomorphism. This is the case also for signed graphs when we use signedhomomorphism. The notion of homomorphism of signed graphs was introducedby Guenin [28], and later systematically studied by Naserasr et al. [47] (cf. alsothe extended abstract [45]). Given two signed graphs (G,Σ1) and (H,Σ2), wesay there is a homomorphism of (G,Σ1) to (H,Σ2) if there is (G,Σ′1), which isequivalent to (G,Σ1), (H,Σ′2), which is equivalent to (H,Σ′2), and a mappingΦ : V (G) → V (H) such that every edge of (G,Σ′1) is mapped to an edge of(H,Σ′2) of the same sign. A signed chromatic number of a signed graph (G,Σ)is the smallest order of a signed graph to which (G,Σ) admits a homomorphism.The signed chromatic number of a balanced signed graph is the same as thechromatic number of an unsigned graph, hence this definition differs from the oneby Zaslavsky. Moreover, it also differs from the one by Macajova, Raspaud andSkoviera, for example, for an unbalanced C4 (the chromatic number of unbalancedC4 is 4 in terms of homomorphism, while it is 3 in the other case). Signedhomomorphisms were investigated by Naserasr et al. [44] (cf. also the extended

18

abstract [46]), Foucaud et al. [26], Brewster et al. [11], Ochem et al. [48], andrecently by Das et al. in [18] and Das et al. in [17].

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