Improved upper bound on the Frank number of 3-edge-connected graphs

In an orientation O of the graph G , an arc e is deletable if and only if O − e is strongly connected. For a 3-edge-connected graph G , the Frank number is the minimum k for which G admits k strongly connected orientations such that for every edge e of G the corresponding arc is deletable in at least one of the k orientations. H¨orsch and Szigeti conjectured the Frank number is at most 3 for every 3-edge-connected graph G . We prove an upper bound of 5, which improves the previous bound of 7.


Introduction
The graphs in this paper are finite and without loops or multiple edges.We recommend the excellent book by Bondy and Murty [2] for the concepts and notations used here.
A graph G is defined by its vertex set V and edge set E. An orientation of G is a directed graph D = (V, A) such that each edge uv ∈ E is replaced by exactly one of the arcs (u, v) or (v, u).A circuit is a directed cycle.A graph is cubic if every vertex has degree 3. A graph is k-edge-connected if and only if the removal of any k − 1 edges leaves a connected graph.A 2-edge-connected graph is often called bridgeless.
A directed graph is strongly connected if and only if selecting two arbitrary vertices x and y, there is a directed (x, y)-path.An orientation of G is k-arc-connected if and only if the removal of any k − 1 arcs leaves a strongly connected directed graph.
Theorem 1.1 (Robbins).A graph has a strongly connected orientation if and only if it is 2-edge-connected.
The following theorem is a fundamental result in the theory of directed graphs [10].
Theorem 1.2 (Nash-Williams).A graph has a k-arc-connected orientation if and only if it is 2k-edge-connected.
These results served as a motivation for Hörsch and Szigeti [7] to investigate the following concepts.In an orientation O of G, the arc e is deletable if and only if O−e is strongly connected.For a 3-edge-connected graph G, the Frank number, denoted by F (G), is the minimum k for which G admits k strongly connected orientations such that for every edge e of G the corresponding arc is deletable in at least one of the k orientations.Why 3-edgeconnected graphs?Suppose G has a cut of size at most 2. In any orientation of G, the removal of any of these edges results in either a directed cut or a directed graph that is not even connected.Hence no set of orientations can satisfy the conditions.On the other hand, if G is 4-edge-connected, then G admits a 2-arc-connected orientation by Theorem 1.2.This orientation yields that F (G) = 1 by definition.Consequently, F (G) = 1 if and only if G is 4-edge-connected.Thus the problem is interesting only if the edge-connectivity of G is 3, i.e. κ ′ (G) = 3.In the sequel, we consider only graphs with edge-connectivity 3.
Hörsch and Szigeti [7] showed that any 3-edge-connected graph G satisfies F (G) ≤ 7. Prior to that, DeVos et al. proved a more general result with a weaker bound [4]: For every 3-edgeconnected graph G, there exists a partition of E(G) into at most nine sets {X 1 , X 2 , . . ., X m } so that G \ X i is 2-edge-connected for every 1 ≤ i ≤ m.Our main result improves the best known upper bound on the Frank number.
Theorem 1.3.For every 3-edge-connected graph the Frank number is at most 5.
The paper is organised as follows.In the second section, we introduce the main tools and results, which we use in our proof.The third section is dedicated to the proof of our main result.We conclude by discussing the limits of our proof technique.

Preliminaries
For some integer k, a k-flow (o, v) on a graph G consists of an orientation o of the edges of G and a valuation v : E(G) → {0, ±1, ±2, . . ., ±(k − 1)} such that at every vertex the sum of the values on incoming edges equals the sum on the outgoing edges.A k-flow (o, v) is nowhere-zero if the value of v is not 0 for any edge of G.A nowhere-zero k-flow on G is all-positive if the value v(e) is positive for every edge e of G. Every nowhere-zero k-flow can be transformed to an all-positive nowhere-zero k-flow by changing the orientation of the edges with negative v(e) and changing negative values of v(e) to −v(e).Inspired by their ideas and approach, we use the following result by Goedgebeur et al. [6].
Lemma 2.1.Let G be a 3-edge-connected graph, and let (o, v) be an all-positive nowhere-zero k-flow on G. Any edge of G, which receive value Indeed, an orientation arising from a flow is always strongly connected, and the removal of any arc of value 1 cannot create a directed cut since the flow is nowhere-zero.Using a slightly stronger Lemma, Goedgebeur et al. [6] proved the following result, which we state without proof.It is well-known that every 3-edge-colorable cubic graph admits a nowhere-zero 4-flow.Tutte posed the following stronger claim, which inspired a vast amount of research.Everyone believes the validity of this conjecture.This partly explains why the only known examples of graphs with Frank number 3 are created from the Petersen graph using certain operations [1].Each of the constructed graphs contains the Petersen graph as a minor.
Since we consider 3-edge-connected graphs only, we can use the following result by Jaeger [8], which was later improved by Seymour [11].
Let H denote an Abelian group.An H-flow on an oriented graph D is an assignment of values of H to the arcs of D such that for each vertex v, the sum of the values on the incoming arcs is the same as the sum of the values on the outgoing arcs.For a graph G, an H-flow is defined accordingly.A nowhere-zero H-flow on G is an H-flow, where 0 ∈ H is not assigned to any edge.The following is a useful corollary of a theorem by Tutte: In what follows, we particularly use a Z 2 × Z 2 × Z 2 -flow on a 3-edge-connected graph G.We use 0/1 vectors with three coordinates to denote elements of Z 2 × Z 2 × Z 2 .The flow condition at a vertex v implies the following property: in each coordinate, the sum of values on edges incident to v is 0. Hence there are an even number of 1's in each coordinate, on the edges incident to a fixed vertex.

Improvement of the upper bound
Surprisingly, Theorem 2.4, the weaker flow result of Jaeger is the useful for our purposes.The main idea of the proof is the following.We fix a nowhere-zero 8-flow of a 3-edge-connected graph G existing by Theorem 2.4.We create 5 other nowhere-zero k-flows of G in such a way that we control the set of edges with value 1, and we apply Lemma 2.1.Since every edge of G receives value 1 in at least one of the flows, we are done.Let us recall our main theorem.
Theorem 1.3 For every 3-edge-connected graph the Frank number is at most 5.
Proof.Combining Theorem 2.4 and Fact 2.6, we consider a nowhere-zero Z 2 × Z 2 × Z 2 -flow on G.For i ∈ {1, 2, 3}, let G i denote the subgraph of G induced by those edges of E, which have value 1 at the ith coordinate.Note that these subgraphs might have some edges in common, more precisely the number of nonzero coordinates of each edge is the same as the number of subgraphs to which it belongs.By the nowhere-zero property, every edge is contained in at least one of these subgraphs.
By the flow condition, the subgraphs G 1 , G 2 , G 3 are Eulerian.We can think of an Eulerian trail as a directed graph.Thus we can partition each edge set E(G i ) into edge-disjoint circuits.We may use some vertices in different Eulerian trails.For i = 1, 2, 3, let us fix an orientation o i of E(G i ).At this point, it might occur that an edge of G has different orientations in different subgraphs G i .The next aim is to select, for each i ∈ {1, 2, 3}, an appropriate positive value v i to send along o i .
Let us emphasize that after fixing the pair (o i , v i ) for each G i , we define another flow (O 1 , f 1 ) in the next step.The orientation and value of an arc in (O 1 , f 1 ) is determined by the superposition of the chosen (o i , v i ) pairs or triples.We always assign a positive value.If the orientations go opposite, then we let the largest value determine the direction and subtract the values going in opposite direction.
Let us construct an all-positive nowhere-zero 8-flow (O 1 , f 1 ) by sending values v 1 = 1, v 2 = 2 and v 3 = 4 along the fixed orientations o i of G i , respectively.Indeed, this is a flow, and there can be no arcs of value 0. The maximum value of an arc is 7, if this edge was of type (1, 1, 1) in the original Z 2 × Z 2 × Z 2 -flow, and the edge received the same orientation in all three orientations o i .Since in every edge-cut, the sum of the flow values in the two directions is the same, there are no directed cuts if we take the superposition of the flows o 1 , o 2 , o 3 .Thus O 1 is strongly connected.
In order to prove F (G) ≤ 5, we create a set of strongly-connected orientations O of G such that |O| = 5, and any edge of G is deletable in at least one of the orientations of O. Let O 1 be the first member of O.By Lemma 2.1, the arcs of value 1 in (O 1 , f 1 ) are deletable with respect to O 1 .What are these arcs?We summarize the possible final orientations and values of the arcs in Table 1.
There are three types of arcs of value 1 in (O 1 , f 1 ).We create 4 other flows such that the larger valued arcs of O 1 receive value 1 in at least one orientation.We refer to any edge of G as one which belongs to the corresponding row of Table 1.The next goal is to create 4 other strongly connected orientations such that every type of edge defined by Table 1 gets flow value 1 in at least one of those orientations, and hence is deletable in those orientations by Lemma 2.1.We achieve this by combining the following two things: we reverse the orientations o i in G i for each i ∈ {1, 2, 3} if necessary, and we change the values v i appropriately.By reversing the orientations, we can switch the role of a fixed edge with respect to its role described in Table 1  Let us continue with some comments and observations on the proof.
Remark 3.1.It does not really matter what kind of flow we use to construct an orientation of O.However, there are two key observations: since the orientation arises from flows, it cannot have a directed cut.Therefore, it is a strongly connected orientation, and by the nowhere-zero property and the 3-edge-connectivity of G an arc of value 1 cannot be the only arc going in the opposite direction in any cut.
Remark 3.2.Note that we do not claim that these are the only deletable edges of O i ∈ O.It may happen that some arcs with higher values are also deletable.Hence our result is slightly stronger than just proving F (G) ≤ 5 since we only relied on the arcs of value 1.

Fact 2 . 6 .
If H and H ′ are two finite Abelian groups of the same order, then the graph G has an H-flow if and only if G has an H ′ -flow.
, and by changing the values we can change the final values of the edges.We define the following two 10-flows (O 2 ,O 3 ) and two 8-flows (O 4 ,O 5 ) based on the orientations o 1 , o 2 , o 3 .Again, if an edge is present in several circuits, we superimpose the values as before 1 to make the flow (O i , f i ) all-positive for all i ∈ {1, 2, 3, 4, 5}.We define O 2 by keeping the same orientations as in O 1 , but sending value 4 along o 1 , value 2 along o 2 , and value 3 along o 3 .The only difference between O 3 and O 2 is that the orientations of the circuits in o 3 are reversed.We get O 4 from O 1 by reversing the orientation of the circuits in o 2 and sending value 2 along o 1 , value 1 along o 2 and value 4 along o 3 .The only difference between O 5 and O 4 is that the values along o 2 and o 3 are swapped.We indicate in Table2which edges receive value 1 in the corresponding orientations of those edges, which did not receive value 1 with 1 the larger value deciding the direction except if 2+3 goes in one direction and 4 in the opposite

Table 1 :
The possible orientations and values of e in (O 1 , f 1 ).