Nowhere-Zero 3-Flows in Signed Graphs Nowhere-Zero 3-Flows in Signed Graphs

. Tutte observed that every nowhere-zero k -ﬂow on a plane graph gives rise to a k - vertex-coloring of its dual, and vice versa. Thus nowhere-zero integer ﬂow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph G has a face-k -colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k -ﬂow. However, if the surface is nonorientable , then a face-k -coloring corresponds to a nowhere-zero k -ﬂow in a signed graph arising from G . Graphs embedded in orientable surfaces are therefore a special case that the corresponding signs are all positive. In this paper, we prove that if an 8-edge-connected signed graph admits a nowhere-zero integer ﬂow, then it has a nowhere-zero 3-ﬂow. Our result extends Thomassen’s 3-ﬂow theorem on 8-edge-connected graphs to the family of all 8-edge-connected signed graphs. And it also improves Zhu’s 3-ﬂow theorem on 11-edge-connected signed graphs.


Introduction.
Graphs considered in this paper may have multiple edges and loops unless otherwise stated.Let G = (V, E) be a graph and let k be a positive integer.An ordered pair (D, f ) is called a k-flow of G if D = (V, A) is an orientation of G and f : A → {0, ±1, . . ., ±(k − 1)} is an assignment of flows, such that, for each v ∈ V , f (e), where E + (v) is the set of all arcs leaving vertex v in D and E − (v) is the set of all arcs entering vertex v.We say that the k-flow (D, f ) is nowhere-zero if f (e) = 0 for any e ∈ A. The concept of nowhere-zero integer flow was introduced by Tutte in 1954, and the theory of integer flows provides an interesting way to extend theorems about region-coloring planar graphs to general graphs [12,13] (see also [15]).Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a k-vertex-coloring of its dual, and vice versa.Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts, and the above Tutte's observation is often referred to as the duality theorem.One of the major open problems in this research area is Tutte's 3-flow conjecture, which is exactly the dual version of Grötzsch's 3-color theorem on planar graphs [3,4].
Thomassen [11] made a breakthrough in this conjecture by establishing the following weaker version.
As proved by Kochol [7], a minimum counterexample to the 3-flow conjecture is 5-edge-connected.Therefore, the above theorem is actually just one step away from the resolution.
The aforementioned duality theorem cannot be extended directly to embedded graphs.(See DeVos et al. [2] for an asymptotic version.)Nevertheless, Jaeger [5] showed that if a graph G has a face-k-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k-flow.Interestingly, if the surface is nonorientable, then this coloring corresponds to a nowhere-zero k-flow in a signed graph arising from G. It is due to their great theoretical interest that integer flows in sign graphs have also been subjects of extensive research.
Let us define a few terms before proceeding.A signed graph is a pair (G, σ), where G is a graph and σ : E(G) → {1, −1} is a signature of G.An edge e is called positive if σ(e) = 1 and negative otherwise.Each edge e = xy of a signed graph, (G, σ) is composed of two half-edges h x and h y , where h x is incident with x and h y is incident with y.An orientation D of (G, σ) assigns every half-edge a direction in the following way: if e = xy is positive, then h x and h y are directed both from x to y, or both from y to x (see Figure 1); if e = xy is negative, then the directions of h x and h y are opposite.(There are two possibilities: (1) h x is directed to x h y is directed to y; (2) h x is directed from x and h y is directed from y. See Figure 1.) A negative edge e = xy is called a source edge if e is directed toward both x and y, and it is called a sink edge otherwise.In the literature, an oriented signed graph is also called a bidirected graph.If all edges of (G, σ) are positive, then a signed graph is equivalent to a graph.So we can view signed graphs as generalizations of graphs.
The concept of nowhere-zero integer flow in graphs carries over naturally to signed graphs, and the following is a well-known conjecture on integer flows in signed graphs.
Despite tremendous research effort, this conjecture remains open; Xu and Zhang [14] confirmed it for 6-edge-connected signed graphs.In [10], Raspaud and Zhu established that every 4-edge-connected signed graph has a nowhere-zero 4-flow provided it admits a nowhere-zero integer flow.Based on Theorem 1.2, Zhu [16] (Zhu [16]).Every 11-edge-connected signed graph admitting a nowhere-zero integer flow has a nowhere-zero 3-flow.
What is the least edge-connectivity that can guarantee the existence of nowherezero 3-flows in signed graphs?Zhu posed this as an open question in [16].With the motivation to improve the bound in Theorem 1.3 and extend the setting of Theorem 1.1, we establish the following main result in this paper.
It is worthwhile pointing out that the assertion no longer holds if 8 is replaced by 4: Let (G, σ) be the signed graph with three vertices in which each pair of vertices is connected by precisely one positive edge and precisely one negative edge.Clearly, G is 4-edge-connected and has a nowhere-zero 4-flow.Nevertheless, it is routine to check that G admits no nowhere-zero 3-flow.

Operations.
In this section we introduce some operations on signed graphs which will be employed in subsequent proofs.
Flipping.Let (G, σ) be a signed graph and let A be a subset of V (G).Define where Ā = V (G) \ A and [A, Ā] is the cut in G consisting of all edges between A and Ā.We say that the signed graph (G, σ ) is obtained from (G, σ) by flipping all edges in [A, Ā].Two signed graphs (G, σ) and (G, σ ) are called equivalent if one can be obtained from the other by flipping all edges in a cut.The following two lemmas are well-known facts (see [10] and [16]) in graph theory, that is, that this flipping operation does not affect the existence of a nowhere-zero integer flow in a signed graph.
Lemma 2.1.Let (G, σ) and (G, σ ) be two equivalent signed graph and let k be a positive integer.Then (G, σ) has a nowhere-zero k-flow if and only if so does (G, σ ).
Throughout we use n(G, σ) to denote the minimum number of negative edges contained in a signed graph equivalent to (G, σ).
Contraction.Let (G, σ) be a signed graph and let A be a subset of V (G).The signed graph obtained from (G, σ) by contracting A, denoted by (G/A, σ), is the graph arising from (G, σ) by identifying all vertices in A to a single vertex, in which each edge of G with both ends in A becomes a loop, and each edge has the same sign as in (G, σ).
Since the sign of a loop is not effected by a flipping operation, the following statement holds.
Lifting  We say that the signed graph (G , σ ) is obtained from (G, σ) by lifting xy and xz; see Figure 2 for an illustration.Note that x, y, z are not necessary distinct in this definition.
An orientation of (G , σ ) can be extended naturally to an orientation of (G, σ) by orienting the two half-edges incident with x as follows: one enters x and the other leaves x; see Figure 2  Let G be a graph and let x, y be two distinct vertices of G.The local edgeconnectivity of G between x and y, denoted by λ G (x, y), is the maximum number of edge-disjoint paths connecting x and y in G.The following Mader's theorem [9] asserts that the local edge-connectivity is preserved under some lifting operation.
Theorem 2.5 (Mader [9]).Let G be a connected loopless graph and let v 0 be a vertex of degree at least 4 such that no edge incident with v 0 is a cut-edge of G. Then G contains two edges v 0 v 1 and v 0 v 2 such that λ H (x, y) = λ G (x, y) for any two vertices x, y different from v 0 , where H is the graph obtained from G by lifting v 0 v 1 and v 0 v 2 .
As shown by Tutte [12], a graph G admits a modulo 3-orientation if and only if it has a nowhere-zero 3-flow; this equivalence relation can be further extended to signed graphs.
The remainder of this paper is devoted to a proof of Theorem 3.

The proof proceeds by induction on |V (G)| + |E(G)|;
to make the induction work, we need a generalized concept of graph orientation and a set function from [8], which is a variant of the one introduced by Thomassen in [11].
Let G be a loopless graph.A mapping β : for all v ∈ V (G).This mapping τ can be further extended to any nonempty A ⊆ V (G) as follows: where β(A) ≡ v∈A β(v) (mod 3).Since d(A) and τ (A) have the same parity, the following inequality holds.Lemma 3.3 (Lovász et al. [8]).
2 is an immediate corollary of the following result, which was derived by refining Thomassen's technique [11] and will be used in our proof.
Theorem 3.4 (Lovász et al. [8]).Let G be a loopless graph, let β be a Z 3boundary of G, let z 0 ∈ V (G), and let D(z 0 ) be a preorientation of the set E(z 0 ) of all edges incident with z 0 .Assume that Then D(z 0 ) can be extended to a β-orientation D of the entire graph G.
When restricted to the disjoint union of an isolated vertex z 0 and a 6-edgeconnected loopless graph, the preceding theorem yields the following statement.
Proof.Let m be the number of negative edges of (G, σ).Set r = 1 if m = 2 and r = 0 if m = 3.Let H be the graph obtained from G by first orienting r negative edges as sink edges and the remaining m − r negative edges as source edges, then inserting a new vertex to each negative edge, and finally identifying all these newly inserted vertices to a single vertex z 0 .Let G = H if m = 2 and let G be obtained from H by replacing one arc leaving z 0 with two parallel arcs entering z 0 if m = 3.
Therefore, by Theorem 3.4, the preorientation of the arcs incident with z 0 can be extended to a modulo 3-orientation of the entire graph G , which clearly yields a modulo 3-orientation of (G, σ).
Lemma 3.7.Let G be a loopless graph, let β be a Z 3 -boundary of G, let z 0 ∈ V (G), let D(z 0 ) be a preorientation of the set E(z 0 ) of all edges incident with z 0 , and let Let p be the integer in Z 3 with β(z 0 ) − d(z 0 ) + 1 ≡ 2p (mod 3) and let q = 7 − d(z 0 )−p.Then q ≥ 0 and p+q ≥ 2 as d(z 0 ) ≤ 5. Let G be obtained from G by adding a set P of p arcs from S to z 0 and adding a set Q of q arcs from z 0 to S such that each vertex in S has degree at least six in G .(This G is available because |S| ≤ 2.) Let β (z 0 ) be the integer in Z 3 with β (z 0 ) ≡ β(z 0 )+q −p (mod 3).By the definitions of p and q, we obtain β (z 0 ) ≡ (d(z 0 )−1+2p)+(7−d(z 0 )−p)−p ≡ 0 (mod 3).So β (z 0 ) = 0.For each vertex v = z 0 , let P (v) (resp., Q(v)) be the set of all arcs in P (resp., Q) incident with v, and let β (v) be the integer in  Assume on the contrary that (G, σ) is a smallest counterexample and, subject to this, the number of negative edges in (G, σ) is minimum.
Let G be the loopless graph (with no signature) obtained from the signed graph (G/ Ā, σ) by first deleting all negative edges and then deleting all loops incident with z 0 , the vertex arising from contracting Ā.We orient all edges between A and z 0 in G as follows: Suppose edge xz 0 in G with x ∈ A corresponds to edge v A y in G with y ∈ Ā.Then the direction of xz 0 in G is exactly the same as the direction of v A y in D .For convenience, we denote this preorientation of edges incident with z 0 by D(z 0 ).Let p(z 0 ) (resp., q(z 0 )) be the number of all resulting arcs entering (resp., leaving) z 0 ; we define β (z 0 ) to be the integer in Z 3 with β (z 0 ) ≡ q(z 0 ) − p(z 0 ) (mod 3).Let F 1 be the set of all negative edges of G with both ends in A. Recall that ( 8) We orient all edges in F 1 as sink edges if k(A, σ) ≡ 2 (mod 3), and orient all edges in F 1 as source edges otherwise.Let F 2 be the set of all negative edges between A and Ā in G; for each edge f ∈ F 2 , we orient it as in D .Set F = F 1 ∪ F 2 .For each v ∈ A, let p(v) (resp., q(v)) be the number of all half-arcs entering (resp., leaving) v in F ; we define β (v) to be the integer in Z 3 with β (v) ≡ p(v) − q(v) (mod 3).We propose to show that (9) β is a Z 3 -boundary of G .
To justify this, let p 1 (resp., q 1 ) be the number of positive edges directed from A to Ā (resp., from Ā to A) in D , and let p 2 (resp., q 2 ) be the number of source (resp., sink) edges between A and Ā in D .Note that (10) p 1 = p(z 0 ) and q 1 = q(z 0 ).
Since d + D (v A ) ≡ d − D (v A ) (mod 3), the following equality holds.

Fig. 2 .
Fig. 2. A lifting of xy and xz and an orientation extension.
for the case when σ(xy) = σ(xz) = −1.Lemma 2.4.Let (G, σ) be a signed graph and let xy, xz be two edges of G.If (G , σ ) is the signed graph obtained from (G, σ) by lifting xy and xz, thenn(G , σ ) ≥ n(G, σ) − 2. Proof.For each U ⊆ V (G), let [U, Ū ] G (resp., [U, Ū ] G )denote the cut consisting of all edges between U and Ū in G (resp., in G).Suppose the signed graph (G , σ ) obtained from (G , σ ) by flipping all edges in a cut [A, Ā] G has precisely n(G , σ ) negative edges.Consider the signed graph (G, σ) obtained from (G, σ) by flipping all edges in [A, Ā] G .It is easy to see that the number of negative edges in (G, σ) is at most two plus the number of negative edges in (G , σ ).Hence, n(G, σ) ≤ n(G , σ ) + 2, as desired.

3 .
Orientations: Modulo and beyond.Let (G, σ) be a signed graph.For each A ⊆ V (G), the degree of A, denoted by d(A), is the number of edges between A and Ā; we write d(A) = d(a) if A = {a}.(Notice that the contribution to d(a) made by any loop incident with a, if any, is zero.)For each orientation D of (G, σ), let d + D (v) (resp., d − D (v)) denote the number of half-arcs leaving (resp., entering) a vertex v; we may drop the subscript D if there is no danger of confusion.Note that, by definition, each loop incident with v (if any) contributes two to d + D (v) + d − D (v), so d(v) < d + D (v) + d − D (v) if such a loop exists.Downloaded 03/17/15 to 147.8.204.164.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php For each A ⊆ V (G ), we use d (A) and τ (A) to denote the degree of A in G and the value of the set function at A, respectively.If m = 2, then d (z 0 ) = 4 ≤ 4 + |τ (z 0 )|.If m = 3, then d (z 0 ) = 7.So |τ (z 0 )| = 3 by definition and thus d (z 0 ) = 4 + |τ (z 0 )|.Hence the inequality d (z 0 ) ≤ 4 + |τ (z 0 )| holds in either case.By Lemma 3.3, we have d then D(z 0 ) can be extended to a β-orientation D of the entire graph G. Proof.By definition, d(z 0 ) and τ (z 0 ) have the same parity, so |τ (z 0 )| ≥ 1 if d(z 0 ) = 5.Hence, d(z 0 ) ≤ 4 + |τ (z 0 )|.If S = ∅, then the statement follows instantly from Theorem 3.4.Thus we may assume S = ∅.
and τ (A) denote the degree of A in G and the value of the set function at A, respectively.Since d (z 0 ) = 7 and β (z 0 ) = 0, we have |τ(z 0 )| = 3.So d (z 0 ) = 4 + |τ (z 0 )|.Since d (v) ≥ 6 for each v ∈ S, from Lemma 3.3 it follows that d (v) ≥ 4 + |τ (v)|.Therefore, by Theorem 3.4, the preorientation of the arcs incident with z 0 can be extended to a β -orientation of the entire graph G , which clearly yields a β-orientation of (G, σ).