aa r X i v : . [ m a t h . C O ] D ec Circular Flows in Planar Graphs
Daniel W. Cranston ∗ Jiaao Li † Abstract
For integers a ≥ b >
0, a circular a/b -flow is a flow that takes values from {± b, ± ( b +1) , . . . , ± ( a − b ) } .The Planar Circular Flow Conjecture states that every 2 k -edge-connected planar graph admits a circular(2 + k )-flow. The cases k = 1 and k = 2 are equivalent to the Four Color Theorem and Gr¨otzsch’s3-Color Theorem. For k ≥
3, the conjecture remains open. Here we make progress when k = 4 and k = 6. We prove that (i) every -edge-connected planar graph admits a circular / -flow and (ii) every -edge-connected planar graph admits a circular / -flow. The dual version of statement (i) oncircular coloring was previously proved by Dvoˇr´ak and Postle (Combinatorica 2017), but our proof hasthe advantages of being much shorter and avoiding the use of computers for case-checking. Further,it has new implications for antisymmetric flows. Statement (ii) is especially interesting because thecounterexamples to Jaeger’s original Circular Flow Conjecture are 12-edge-connected nonplanar graphsthat admit no circular 7 / For integers a ≥ b >
0, a circular a/b -flow circular a/b -flow1 is a flow that takes values from {± b, ± ( b + 1) , . . . , ± ( a − b ) } .In this paper we study the following conjecture, which arises from Jaeger’s Circular Flow Conjecture [9]. Conjecture 1.1 (Planar Circular Flow Conjecture) . Every k -edge-connected planar graph admits a circular (2 + k ) -flow. When k = 1 this conjecture is the flow version of the 4 Color Theorem. It is true for planar graphs (by4CT), but false for nonplanar graphs because of the Petersen graph, and all other snarks. Tutte’s 4-FlowConjecture, from 1966, claims that Conjecture 1.1 extends to every graph with no Petersen minor. When k = 2, Conjecture 1.1 is the dual of Gr¨otzsch’s 3-Color Theorem. Tutte’s 3-Flow Conjecture, from 1972,asserts that it extends to all graphs (both planar and nonplanar). In 1981 Jaeger further extended Tutte’sFlow Conjectures, by proposing a general Circular Flow Conjecture: for each even integer k ≥ , every k -edge-connected graph admits a circular (2 + k ) -flow . That is, he believed Conjecture 1.1 extends to allgraphs for all even k . A weaker version of Jaeger’s conjecture was proved by Thomassen [17], for graphs withedge connectivity at least 2 k + k . This edge connectivity condition was substantially improved by Lov´asz,Thomassen, Wu, Zhang [13]. Theorem 1.2. (Lov´asz, Thomassen, Wu, Zhang [13])
For each even integer k ≥ , every k -edge-connectedgraph admits a circular (2 + k ) -flow. In contrast, Jaeger’s Circular Flow Conjecture was recently disproved for all k ≥
6. In [8], for each eveninteger k ≥
6, the authors construct a 2 k -edge-connected nonplanar graph admitting no circular (2 + k )-flow. ∗ Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA; [email protected] ; This research is partially supported by NSA Grant H98230-15-1-0013. † School of Mathematical Sciences and LPMC Nankai University, Tianjin 300071, China; [email protected] Jaeger [9] showed that if p, q, r, s ∈ Z + and p/q = r/s , then each graph G has a circular p/q -flow if and only if it has acircular r/s -flow. (See [7] for more details.) We use this result implicitly in the present paper. k , we can also modify the construction in [8] to get 2 k -edge-connected nonplanargraphs admitting no circular (2 + k )-flow. Thus, the planarity hypothesis of Conjecture 1.1 seems essential.The case k = 4 of Jaeger’s Circular Flow Conjecture, which remains open, is particularly important, sinceJaeger [9] observed that if every 9-edge-connected graph admits a circular 5 / k ∈ { , } . Theorem 1.3.
Every -edge-connected planar graph admits a circular / -flow. Theorem 1.4.
Every -edge-connected planar graph admits a circular / -flow. The dual version of Theorem 1.3, on circular coloring, was proved by Dvoˇr´ak and Postle [5]. In fact,their coloring result holds for a larger class of graphs that includes some sparse nonplanar graphs, as wellas all planar graphs with girth at least 10. However, our proof is much shorter and avoids using computersfor case-checking. Our proof also has new implications for antisymmetric flows (see Theorem 2.4 below).Theorem 1.4 is especially interesting because the counterexamples in [8] to Jaeger’s original circular flowconjecture are 12-edge-connected nonplanar graphs that admit no circular 7 / Graphs in this paper are finite and can have multiple edges, but no loops. Our notation is mainly standard.For a graph G , we write | G | for | V ( G ) | and write k G k for | E ( G ) | | G | , k G k . Let δ ( G ) denote the minimum degree ina graph G . A k -vertex is a vertex of degree k . For disjoint vertex subsets X and Y , let [ X, Y ] G [ X, Y ] G denote theset of edges in G with one endpoint in each of X and Y . Let X c = V ( G ) \ X X c , d ( X ) , and let d ( X ) = | [ X, X c ] | . Forvertices v and w , let µ ( vw ) = | [ { v } , { w } ] G | and µ ( G ) = max v,w ∈ V ( G ) µ ( vw ) µ ( vw ) , µ ( G ) .To lift lift a pair of edges w v , vw incident to a vertex v in a graph G means to delete w v and vw andcreate a new edge w w . To contract contract an edge e in G means to identify its two endpoints and then delete theresulting loop. For a subgraph H of G , we write G/H to denote the graph formed from G by successivelycontracting the edges of E ( H ). The lifting and contraction operations are used frequently in this paper.An orientation D of a graph G is a modulo (2 p + 1) -orientation modulo(2 p + 1)-orientation if d + D ( v ) − d − D ( v ) ≡ p + 1) foreach v ∈ V ( G ). By the following lemma of Jaeger [9], this problem is equivalent to finding circular flows (fora short proof, see [19, Theorem 9.2.3]). Lemma 1.5. [9] A graph admits a circular (2 + p ) -flow if and only if it has a modulo (2 p + 1) -orientation. To prove our results, we study modulo orientations. Let G be a graph. A function β : V ( G ) Z p +1 is a Z p +1 -boundary Z p +1 -boundary if P v ∈ V ( G ) β ( v ) ≡ p + 1). Given a Z p +1 -boundary β , a ( Z p +1 , β ) -orientation ( Z p +1 , β )-orientation isan orientation D such that d + D ( v ) − d − D ( v ) ≡ β ( v ) (mod 2 p + 1) for each v ∈ V ( G ). When such an orientationexists, we say that the boundary β is achievable achievable . If β ( v ) = 0 for all v ∈ V ( G ), then a ( Z p +1 , β )-orientation issimply a modulo (2 p + 1)-orientation. As defined in [10, 11], a graph G is strongly Z p +1 -connected strongly Z p +1 -connected if for any Z p +1 -boundary β , graph G admits a ( Z p +1 , β )-orientation. When the context is clear, we may simply write β -orientation β -orientation for ( Z p +1 , β )-orientation. Suppose we are given a graph G , an integer p , a Z p +1 -boundary β for G , and a connected subgraph H ( G . We form G ′ from G by contracting H ; that is G ′ = G/H . Let w denote the new vertex in G ′ , formed by contracting E ( H ). Define β ′ for G ′ by β ′ ( v ) = β ( v ) for each v ∈ V ( G ′ ) \ { w } , and β ′ ( w ) = P v ∈ V ( H ) β ( v ) (mod 2 p + 1). Note that β ′ is a Z p +1 -boundary for G ′ . Themotivation for generalizing modulo orientations is the following observation of Lai [10], which is also appliedin Thomassen et al. [17, 13]. Lemma 1.6 ([10]) . Let G be a graph with a subgraph H , and let G ′ = G/H . Let β and β ′ be Z p +1 boundaries(respectively) of G and G ′ , as defined above. If H is strongly Z p +1 -connected, then every β ′ -orientation of G ′ can be extended to a β -orientation of G . In particular, each of the following holds.(i) If H is strongly Z p +1 -connected and G/H has a modulo (2 p + 1) -orientation, then G has a modulo (2 p + 1) -orientation. ii) If H and G/H are strongly Z p +1 -connected, then G is also strongly Z p +1 -connected.Proof. We prove the first statement, since it implies (i) and (ii). Fix a β ′ -orientation of G ′ . This yields anorientation D of the subgraph G − E ( G [ V ( H )]). By orienting arbitrarily each edge in E ( G [ V ( H )]) \ E ( H ),we obtain a β ′′ -orientation D of G − E ( H ), for some β ′′ . For each v ∈ V ( H ), let γ ( v ) = β ( v ) − β ′′ ( v ). It iseasy to check that γ is a Z p +1 -boundary of H . Since H is strongly Z p +1 -connected, H has a γ -orientation D . Hence D ∪ D is a β -orientation of G . Proof Outline for Main Results.
To prove Theorems 1.3 and 1.4, we actually establish two stronger,more technical results on orientations; namely, we prove Theorems 2.2 and 3.3. Lemma 1.6 shows thatstrongly Z p +1 -connected graphs are contractible configurations when we are looking for modulo orientations.To prove Theorems 2.2 and 3.3, we use lifting and contraction operations to find many more reducibleconfigurations. These configurations eventually facilitate a discharging proof. The proofs of Theorems 1.3and 1.4 are similar, though the latter is harder. In the next section we just discuss Theorem 1.3, but mostof the key ideas are reused in the proof of Theorem 1.4. / -flows: Proof of Theorem 1.3 -Orientations and Antisymmetric Z -flows To prove Theorem 1.3, we will first present a more technical result, Theorem 2.2, which yields Theorem 1.3as an easy corollary (as we show below in Theorem 2.5). The hypothesis in Theorem 2.2 uses a weightfunction w , which is motivated by the following Spanning Tree Packing Theorem of Nash-Williams [14] andTutte [18]: a graph G has k edge-disjoint spanning trees if and only if every partition P = { P , P , . . . , P t } satisfies P ti =1 d ( P i ) − k ( t − ≥ . This condition is necessary, since in a partition with t parts, eachspanning tree has at least t − G is strongly Z p +1 -connected, then it contains 2 p edge-disjoint spanning trees (although this necessary condition is notalways sufficient). To capture this idea, we define the following weight function. Definition 2.1.
Let P = { P , P , . . . , P t } be a partition of V ( G ). Let w G ( P ) = t X i =1 d ( P i ) − t + 19and w ( G ) = min { w G ( P ) : P is a partition of V ( G ) } . Let T a,b,c T a,b,c denote a 3-vertex graph (triangle) with its pairs of vertices joined by a , b , and c paralleledges; let aH aH denote the graph formed from H by replacing each edge with a parallel edges. For example, w (3 K ) = 3, w (2 K ) = 1, w ( T , , ) = w ( T , , ) = 0; see Figure 1. For each of these four graphs the minimumin the definition of w ( G ) is attained only by the partition with each vertex in its own part. We typicallyassume V ( T a,b,c ) = { v , v , v } and d ( v ) ≤ d ( v ) ≤ d ( v ).Let T = { K , K , T , , , T , , } . Each graph G ∈ T (see Figure 1) is not strongly Z -connected, sincethere exists some Z -boundary β for which G has no β -orientation. A short case analysis shows that noneof the following boundaries are achievable. For 3 K , let β ( v ) = β ( v ) = 0. For 2 K , let β ( v ) = 1 and β ( v ) = 4. For T , , , let β ( v ) = 1 and β ( v ) = β ( v ) = 2. For T , , , let β ( v ) = β ( v ) = 1 and β ( v ) = 3.Now suppose that G has a partition P such that G/ P ∈ T , where the vertices in each P i are identifiedto form v i . To construct a Z -boundary γ for which G has no γ -orientation, we assign boundary γ so that P v ∈ P i γ ( v ) ≡ β ( v i ). Hence G has no γ -orientation precisely because G/ P has no β -orientation. We call apartition P troublesome trouble-some if G/ P ∈ T = { K , K , T , , , T , , } . The main result of Section 2 is Theorem 2.2. Theorem 2.2.
Let G be a planar graph and β be a Z -boundary of G . If w ( G ) ≥ , then G admits a ( Z , β ) -orientation, unless G has a troublesome partition. t K t t K t tt T , , t tt T , , Figure 1: The graphs 3 K , K , T , , , T , , . Before proving Theorem 1.3, we prove a slightly weaker result, assuming the truth of Theorem 2.2.
Theorem 2.3. If G is an 11-edge-connected planar graph, then G is strongly Z -connected.Proof. Let G be an 11-edge-connected planar graph. Fix a partition P . Since G is 11-edge-connected, d ( P i ) ≥
11 for each i , which implies w G ( P ) ≥
19. Thus w ( G ) ≥
19. Since it is easy to see each troublesomepartition P has w ( G/ P ) ≤
3, we obtain that G has no partition P such that G/ P is troublesome. NowTheorem 2.2 implies that G is strongly Z -connected.An antisymmetric Z -flow antisym-metric Z -flow in a directed graph D = D ( G ) is a Z -flow such that no two edges have flowvalues summing to 0. One example is any Z -flow that uses only values 1 and 2. Esperet, de Verclos, Le,and Thomass´e [6] proved that if a graph G is strongly Z -connected, then every orientation D ( G ) of G admits an antisymmetric Z -flow. Together with work of Lov´asz et al. [13], this implies that every directed12-edge-connected graph admits an antisymmetric Z -flow. Esperet et al. [6] conjectured the stronger resultthat every directed -edge-connected graph admits an antisymmetric Z -flow . The concept of antisymmetricflows and its dual, homomorphisms to oriented graphs, were introduced by Neˇsetˇril and Raspaud [16]. In[15], Neˇsetˇril, Raspaud and Sopena showed that every orientation of a planar graph of girth at least 16 hasa homomorphism to an oriented simple graph on at most 5 vertices. The girth condition is reduced to 14 in[4], to 13 in [3], and finally to 12 in [2]. By duality, the results of [16], [6], and [13] combine to imply thatgirth 12 suffices. After the girth 12 result of Borodin et al. [2] in 2007, Esperet et al. [6] remarked that “itis not known whether the same holds for planar graphs of girth at least 11.” Note that the result of Dvoˇr´akand Postle [5] does not seem to apply to homomorphisms to oriented graphs. By Theorem 2.3, we improvethis girth bound for planar graphs. Theorem 2.4.
Every directed -edge-connected planar graph admits an antisymmetric Z -flow. Dually,every orientation of a planar graph of girth at least has a homomorphism to an oriented simple graph onat most 5 vertices. A graph G has odd edge-connectivity odd edge-connectivity t if the smallest edge cut of odd size has size t . Our strongest resulton modulo 5-orientations is the following, which includes Theorem 1.3 as a special case. Theorem 2.5.
Every odd- -edge-connected planar graph admits a modulo 5-orientation. In particular,every 10-edge-connected planar graph admits a modulo 5-orientation (and thus a circular 5/2-flow).Proof. The second statement follows from the first, by Lemma 1.5. To prove the first, suppose the theoremis false, and let G be a counterexample minimizing k G k . By Zhang’s Splitting Lemma for odd edge-connectivity [20], we know δ ( G ) ≥
11. If G is 11-edge-connected, then we are done by Theorem 2.3; soassume it is not. Choose a smallest set W ⊂ V ( G ) such that d ( W ) <
11. Note that | W | ≥
2, and every This says that if G has a vertex v with d ( v ) / ∈ { , } , then we can lift a pair of edges incident to v that are successive inthe circular order around v , and the resulting graph is still planar and odd-11-edge-connected. For example, if d ( v ) = 10, thenall edges incident to v will be lifted in pairs, so the boundary value at v in the resulting orientation will be 0. This is why theproof yields a modulo 5-orientation, but does not show that G is strongly Z -connected. W ′ ( W satisfies d ( W ′ ) ≥
11. Let H = G [ W ]. For any partition P = { P , P , . . . , P t } of H with t ≥
2, we know that d G ( P i ) ≥
11 by the minimality of W , since P i ( W . This implies w H ( P ) = t X i =1 d H ( P i ) − t + 19= t X i =1 d G ( P i ) − d G ( W c ) − t + 19 > t − − t + 19 ≥ . Thus w ( H ) ≥
9, which implies H is strongly Z -connected by Theorem 2.2. By the minimality of G , thegraph G/H has a modulo 5-orientation. By Lemma 1.6, this extends to a modulo 5-orientation of G , whichcompletes the proof. To prove Theorem 2.2, we assume the result is false and study a minimal counterexample. In the nextsubsection we prove many structural results about the minimal counterexample, which ultimately imply itcannot exist. In this subsection we prove that a few small graphs cannot appear as subgraphs of the minimalcounterexample. We call such a forbidden subgraph reducible reducible . By Lemma 1.6, to show that H is reducibleit suffices to show H is strongly Z -connected.Let G be a graph. We often lift a pair of edges w v , vw incident to a vertex v in G to form a new graph G ′ . That is, we delete w v and vw and create a new edge w w . If G ′ is strongly Z k -connected, then so is G , since from any β -orientation of G ′ we delete the edge w w and add the directed edges w v and vw toobtain a β -orientation of G . To prove G is strongly Z k -connected, we use lifting in two similar ways.First, we lift some edge pairs to create a G ′ that contains a strongly Z k -connected subgraph H . If G ′ /H is strongly Z k -connected, then so is G ′ by Lemma 1.6. As discussed in the previous paragraph, so is G .Second, given a Z k -boundary β , we orient some edges incident to a vertex v to achieve β ( v ). For each edge vw that we orient, we increase or decrease by 1 the value of β ( w ). Now we delete v and all oriented edges,and lift the remaining edges incident to v (in pairs). Call the resulting graph and boundary G ′ and β ′ . If G ′ has a β ′ -orientation, then G has a β -orientation. We call these lifting reductions of the first and secondtype liftingreductionsof the firstand secondtype , respectively. In this paper whenever we lift an edge pair vw , wx we require that edge vx already exists.Thus, our lifting reductions always preserve planarity. Lemma 2.6.
Each of the graphs K , T , , , K , and C , shown in Figure 2, is strongly Z -connected. t t K t tt T , , t ttt K C t tt t Figure 2: The graphs 4 K , T , , , K , C . Proof.
Proving the lemma amounts to checking a finite list of cases. So our goal is to make this as painlessas possible. Throughout we fix a Z -boundary β and construct an orientation that achieves β .Let G = 4 K and V ( G ) = { v , v } . To achieve β ( v ) ∈ { , , , , } the number of edges we orient outof v is (respectively) 2, 0, 3, 1, 4. 5et G = T , , and V ( G ) = { v , v , v } , with d ( v ) = d ( v ) = 5 and d ( v ) = 6. If β ( v ) = 0, then weachieve β by orienting 3 edges incident to v , and lifting a pair of unused, nonparallel, edges incident to v to create a fourth edge v v . Since 4 K is strongly Z -connected, we can use the resulting 4 edges to achieve β ( v ) and β ( v ). (This is a lifting reduction of the second type. In what follows, we are less explicit aboutsuch descriptions.) So we assume that β ( v ) = 0 and, by symmetry, β ( v ) = 0. This implies β ( v ) = 0. Nowwe orient all edges from v to v , from v to v and from v to v .Let G = 2 K and V ( G ) = { v , v , v , v } . If β ( v ) ∈ { , , } , then we achieve β ( v ) by orienting twononparallel edges incident to v . Now we lift two pairs of unused edges incident to v to get a T , , . Since T , , is strongly Z -connected, we are done by Lemma 1.6. So assume β ( v ) / ∈ { , , } . By symmetry, weassume β ( v i ) ∈ { , } for all i . Since β is a Z -boundary, we further assume β ( v i ) = 1 when i ∈ { , } and β ( v j ) = 4 when j ∈ { , } . Let V = { v , v } and V = { v , v } . Orient all edges from V to V . For eachpair of parallel edges within V or V , orient one edge in each direction. This achieves β .Let G = 3 C and V ( G ) = { v , v , v , v } with v , v ∈ N ( v ) ∩ N ( v ). If β ( v ) ∈ { , , } , then weachieve β ( v ) by orienting two nonparallel edges incident to v and lifting two pairs of edges incident to v .The resulting unoriented graph is T , , , so we are done by Lemma 1.6. Assume instead, by symmetry, that β ( v i ) ∈ { , } for all i . Since β is a Z -boundary, two vertices v i have β ( v i ) = 1 and two vertices v j have β ( v j ) = 4. By symmetry, assume β ( v ) = 1. If β ( v ) = 1, then orient all edges out from v and v . Assumeinstead, by symmetry, that β ( v ) = 1; now reverse one edge v v from the previous orientation. Definition 2.7.
For partitions P = { P , P , . . . , P t } and P ′ = { P ′ , P ′ , . . . , P ′ s } , we say that P ′ is a refine-ment refinement of P , denoted by P ′ (cid:22) P , if P ′ is obtained from P by further partitioning P i into smaller sets for some P i ’s in P . More formally, we require that for every P ′ j ∈ P ′ , there exists P i ∈ P such that P ′ j ⊆ P i .Since partitions are central to our theorems and proofs, we name a few common types of them. A partition P = { P , P , . . . , P t } is trivial trivial if each part P i is a singleton, i.e., V ( G ) is partitioned into | G | parts; otherwise P is nontrivial nontrivial . A trivial partition is minimal under the relation ≺ . A partition P = { P , P , . . . , P t } is almost trivial almosttrivial if t = | G | − P i with | P i | = 2. A partition P is called normal normal if itis neither trivial nor almost trivial and P 6 = { V ( G ) } .Given a partition P of V ( G ) and a partition Q of G [ P ], the following lemma relates the weights of P , Q , and the refinement Q ∪ ( P \ { P } ). Lemma 2.8.
Let P = { P , P , . . . , P t } be a partition of V ( G ) with | P | > . Let H = G [ P ] and let Q = { Q , Q , . . . , Q s } be a partition of V ( H ) . Now Q ∪ ( P \ { P } ) is a refinement of P satisfying w G ( Q ∪ ( P \ { P } )) = w H ( Q ) + w G ( P ) − . (1) Proof.
Clearly,
Q ∪ ( P \ { P } ) is a refinement of P , and it follows from Definition 2.1 that w G ( Q ∪ ( P \ { P } )) = s X i =1 d G ( Q i ) + t X j =2 d G ( P j ) − s + t −
1) + 19= [ s X i =1 d G ( Q i ) − d G ( P ) − s + 19] + [ t X j =1 d G ( P j ) − t − s X i =1 d H ( Q i ) − s + 19] + [ t X j =1 d G ( P j ) − t + 19] − (19 − w H ( Q ) + w G ( P ) − (19 − . Let G be a counterexample to Theorem 2.2 that minimizes | G | + k G k . Thus Theorem 2.2 holds forall graphs smaller than G . This implies the following lemma, which we will use frequently.6 emma 2.9. If H is a planar graph with w ( H ) ≥ and | H | + k H k < | G | + k G k , then each of the followingholds.(a) If w H ( P ) ≥ for every nontrivial partition P , then H is strongly Z -connected unless H ∈ { K , K , T , , , T , , } .(b) If w ( H ) ≥ and H is -edge-connected, then H is strongly Z -connected.(c) If w ( H ) ≥ , then H is strongly Z -connected.Proof. To prove each part, we fix a Z -boundary β and apply Theorem 2.2 to H . Notice that each troublesomepartition P satisfies w ( G/ P ) ≤
3. So for (a), only the trivial partition can be troublesome. Thus, H isstrongly Z -connected unless H ∈ { K , K , T , , , T , , } . For (b), G has no partition P with G/ P ∈{ K , K } since G is 4-edge-connected. And G has no partition P with G/ P ∈ { T , , , T , , } since w ( H ) ≥
1. So H is again strongly Z -connected, by Theorem 2.2. Finally, (c) follows from (b), since if H has an edge cut [ X, X c ] of size at most 3, then w H ( { X, X c } ) ≤ − w ( H ) ≥ w G ( P ) is relatively largefor each nontrivial partition P . This enables us to slightly modify certain proper subgraphs and still applyLemma 2.9 to the resulting graph H . This added flexibility (to slightly modify the subgraph) helps usto prove that more subgraphs are reducible. In the next section, these forbidden subgraphs facilitate adischarging proof that shows that our minimal counterexample G cannot exist. Claim 1. G has no strongly Z -connected subgraph H with | H | > . In particular,(a) G has no copy of K , T , , , K , or C (by Lemma 2.6), and(b) | G | ≥ .Proof. Suppose to the contrary that H is a strongly Z -connected subgraph of G with | H | >
1, and let G ′ = G/H . Since G is a minimal counterexample, G ′ is strongly Z -connected, by Theorem 2.2. SoLemma 1.6 implies G is strongly Z -connected, which is a contradiction. This proves both the first statementand (a). For (b), clearly | G | ≥
3, since w ( G ) ≥ G / ∈ { K , K } and G contains no 4 K . So assume | G | = 3. Since w ( G/ P ) ≥ P , we know that k G k ≥
8. Since
G / ∈ { T , , , T , , } ,either G contains 4 K or G contains T , , . Each case contradicts (a). Claim 2.
Let P = { P , P , . . . , P t } be a nontrivial partition of V ( G ) . Now(a) w G ( P ) ≥ , and(b) w G ( P ) ≥ if P is normal.Proof. Our proof is by contradiction. For an almost trivial partition P , we have w G ( P ) ≥ w G ( V ( G )) − ≥
5, since G does not contain 4 K by Claim 1(a). If P = { V ( G ) } , then w G ( P ) = 0 −
11 + 19 = 8.Since | G | ≥ P = { P , P , . . . , P t } be a normal partition of V ( G ). By symmetry we assume | P | > H = G [ P ]. For any partition Q = { Q , Q , . . . , Q s } of V ( H ), by Eq. (1) the refinement Q ∪ ( P \ { P } ) of P satisfies w H ( Q ) = w G ( Q ∪ ( P \ { P } )) − w G ( P ) + 8 . (2)(a) We first show that w G ( P ) ≥
5. If w G ( P ) ≤
4, then Eq. (2) implies w H ( Q ) ≥ Q of H , since w G ( Q ∪ ( P \ { P } )) ≥
0. Hence w ( H ) ≥ H is strongly Z -connected by Lemma 2.9(c),which contradicts Claim 1. This proves (a).(b) We now show that w G ( P ) ≥
8. Suppose to the contrary that w G ( P ) ≤
7. If P contains at least twonontrivial parts, say | P | >
1, then (a) implies w G ( Q ∪ ( P \ { P } )) ≥ Q of H . Hence7 ( H ) ≥ H is strongly Z -connected by Lemma 2.9(c), which contradicts Claim 1. Soassume instead that P contains a unique nontrivial part P and | P | ≥
3. For any nontrivial partition Q of H , the refinement Q ∪ ( P \ { P } ) of P is a nontrivial partition of G , and so w G ( Q ∪ ( P \ { P } )) ≥ w H ( Q ) ≥ Q of H by Eq. (2). For the trivial partition Q ∗ of H , since w G ( P ) ≤
7, Eq. (2) implies w H ( Q ∗ ) ≥
1. Since | H | = | P | ≥
3, we know
H / ∈ { K , K } . Since w ( H ) ≥ H = T a,b,c with a + b + c ≤
7. So Lemma 2.9(a) implies that H is strongly Z -connected, whichcontradicts Claim 1.The next two claims are consequences of Claim 2; they give lower bounds on the edge-connectivity of G . Claim 3.
For a partition P = { P , P , . . . , P t } ,(a) if | P | ≥ and | P | ≥ , then w ( P ) ≥ ; and(b) if | P | ≥ and | P | ≥ , then w ( P ) ≥ .Proof. Let H = G [ P ] and Q = { Q , Q , . . . , Q s } be a partition of H . Let P ′ = Q ∪ ( P \ { P } ). Note thatif | P | ≥
2, then the refinement P ′ is nontrivial, and if | P | ≥
3, then P ′ is normal. By Eq. (1), w G ( P ′ ) = w H ( Q ) + w G ( P ) − . (a) If w G ( P ) ≤
9, then w H ( Q ) ≥ Q of H since w G ( P ′ ) ≥ H isstrongly Z -connected by Lemma 2.9(c), which contradicts Claim 1.(b) Similar to (a), if w G ( P ) ≤
12, then w H ( Q ) ≥ Q of H since w G ( P ′ ) ≥ H is strongly Z -connected by Lemma 2.9(c), which contradicts Claim 1. Claim 4.
Let [ X, X c ] be an edge cut of G .(a) Now | [ X, X c ] | ≥ . That is, G is -edge-connected.(b) If | X | ≥ and | X c | ≥ , then | [ X, X c ] | ≥ .Proof. If [
X, X c ] is an edge cut of G , then P = { X, X c } is a partition of V ( G ). (a) Clearly P is normal, since | G | ≥ ≤ w G ( P ) = 2 | [ X, X c ] |−
22 + 19, which yields | [ X, X c ] | ≥ | X | ≥ | X c | ≥
3, then w ( P ) ≥
13 by Claim 3(b). So 13 ≤ w G ( P ) = 2 | [ X, X c ] | −
22 + 19, whichimplies | [ X, X c ] | ≥ G contains no copy of any graph in Figure 3 below. We write H ◦ H ◦ , H ◦◦ to denote the graphformed from H by subdividing one copy of an edge of maximum multiplicity. So, for example, 4 K ◦ = T , , .We write H ◦◦ to denote ( H ◦ ) ◦ . (The reader may think of the ◦ as representing the new 2-vertex.) Claim 5. G has no copy of T , , .Proof. Suppose G contains a copy of T , , , with vertices x, y, z and µ ( xy ) = 3. We lift xz, zy to becomea new edge xy and then contract the corresponding 4 K (contract xy ). Let G ′ denote the resulting graph.The trivial partition Q ∗ of G ′ satisfies w G ′ ( Q ∗ ) ≥ w ( G ) − ≥
1. Every nontrivial partition Q ′ of G ′ corresponds to a normal partition Q of G in which the contracted vertex is replaced by { x, y } . Since xz, zy are the only two edges possibly counted in w G ( Q ) but not in w G ′ ( Q ′ ), we have w G ′ ( Q ′ ) ≥ w G ( Q ) − ≥ w ( G ′ ) ≥
1. By Claim 4, G is 6-edge-connected, so G ′ is 4-edge-connected. Thus G ′ is strongly Z -connected, by Lemma 2.9(b). This is a lifting reduction of the first type, so G is strongly Z -connected, which is a contradiction. Claim 6. G has no copy of C ◦ . tt t tt tt t tt tt T , , C ◦ T ◦◦ , , Figure 3: The graphs T , , , C ◦ , T ◦◦ , , . Proof.
Suppose G contains a copy of 3 C ◦ , with vertices v , v , v , v , z , where z is a 2-vertex with N ( z ) = { v , v } . We lift v z, zv to become a new edge v v and then contract the corresponding 3 C to obtainthe graph G ′ . For the trivial partition Q ∗ of G ′ , we have w G ′ ( Q ∗ ) ≥ w ( G ) − ≥
7. For everynontrivial partition Q ′ of G ′ , we have w G ′ ( Q ′ ) ≥ w G ( Q ) − ≥ w ( G ′ ) ≥
4, so G ′ is strongly Z -connected by Lemma 2.9(c). This is a lifting reduction of thefirst type. Hence G is strongly Z -connected, which contradicts Claim 1.Now we can slightly strengthen Claim 2(b). Claim 7.
Every normal partition P = { P , P , . . . , P t } satisfies w ( P ) ≥ . Proof.
Let P = { P , P , . . . , P t } be a normal partition of G with | P | >
1. Suppose to the contrary that w ( P ) = 8, by Claim 2(b). Now | P | ≥ | P | = . . . = | P t | = 1, by Claim 3(a). As in Claim 2, let H = G [ P ], let Q = { Q , Q , . . . , Q s } be a partition of H , and let P ′ = Q ∪ ( P \ { P } ) be a refinement of P .Eq. (1) implies w H ( Q ) = w G ( P ′ ) − w G ( P ) + 8 = w G ( P ′ ) . If Q is a nontrivial partition of H , then P ′ is nontrivial in G , so w H ( Q ) = w G ( P ′ ) ≥
5, by Claim 2(a). If Q is the trivial partition of H , then w H ( Q ) = w G ( P ′ ) ≥
0. Since | H | = | P | ≥
3, we know
H / ∈ { K , K } .And since G has no copy of T , , , by Claim 5, we know H / ∈ { T , , , T , , } . Now Lemma 2.9(a) impliesthat H is strongly Z -connected, which contradicts Claim 1.Claim 7 allows us to also prove that the third graph in Figure 3 is reducible. Claim 8. G has no copy of T ◦◦ , , .Proof. Suppose G contains a copy of T ◦◦ , , with vertices w, x, y, z , z , where z and z are 2-vertices with N ( z ) = { w, x } and N ( z ) = { x, y } . We lift wz , z x to become a new edge wx , and lift xz , z y to becomea new edge xy . Now { w, x, y } induces a copy of T , , , so we contract { w, x, y } to form a graph G ′ . Since δ ( G ) ≥ δ ( G ′ ) ≥
4. The size of each edge cut decreases at most 4 from G to G ′ , andit decreases at least 3 only if that edge cut has at least two vertices on each side. In that case Claim 4(b)shows the original edge cut in G has size at least 8. Since G is 6-edge-connected by Claim 4, each edge cutin G ′ has size at least 4, so G ′ is 4-edge-connected.The trivial partition Q ∗ of G ′ satisfies w G ′ ( Q ∗ ) ≥ w ( G ) − ≥
2. Every nontrivial partition Q ′ of G ′ corresponds to a normal partition Q of G in which the contracted vertex is replaced by { w, x, y } . So w G ′ ( Q ′ ) ≥ w G ( Q ) − ≥
1, by Claim 7. Thus, G ′ is 4-edge-connected and w ( G ′ ) ≥
1. By Lemma 2.9(b), G ′ is strongly Z -connected. This is a lifting reduction of the first type. Since T , , is strongly Z -connectedby Lemma 2.6, graph G is strongly Z -connected, which contradicts Claim 1.9 .4 The final step: Discharging Now we use discharging to show that some subgraph in Figure 2 or 3 must appear in G . This contradictsone of the claims in the previous section, and thus finishes the proof.Fix a plane embedding of G . (We assume that all parallel edges between two vertices v and w areembedded consecutively, in the cyclic orders, around both v and w .) Let F ( G ) denote the set of all facesof G . For each face f ∈ F ( G ), we write ℓ ( f ) for its length. A face f is a k -face , k + -face , or k − -face k/k + /k − -face if(respectively) ℓ ( f ) = k , ℓ ( f ) ≥ k , or ℓ ( f ) ≤ k . A sequence of faces f f . . . f s is called a face chain face chain if, foreach i ∈ { , . . . , s − } , faces f i and f i +1 are adjacent, i.e., their boundaries share a common edge. The length of this chain is s + 1. Two faces f and f ′ are weakly adjacent weaklyadjacent if there is a face chain f f . . . f s f ′ suchthat that f i is a 2-face for each i ∈ { , . . . , s } . We allow s to be 0, meaning f and f ′ are adjacent. A string string is a maximal face chain such that each of its faces is a 2-face. The boundary of a string consists of twoedges, each of which is incident to a 3 + -face. A k -face is called a ( t , t , . . . , t k )-face if its boundary edges arecontained in strings with lengths t , t , . . . , t k . Here t i is allowed to be 1, meaning the corresponding edge isnot contained in a string.Since w ( G ) ≥
0, we have 2 k G k − | G | + 19 ≥ . By Euler’s Formula, | G | + | F ( G ) | − k G k = 2. We solvefor | G | in the equation and substitute into the inequality, which gives X f ∈ F ( G ) ℓ ( f ) = 2 k G k ≤ | F ( G ) | − . (3)We assign to each face f initial charge ℓ ( f ). So the total charge is strictly less than 22 | F ( G ) | /
9. Toredistribute charge, we use the following three discharging rules.(R1) Each 2-face receives charge from each weakly adjacent 3 + -face.(R2) Each (2 , , from each weakly adjacent 4 + -face and (2 , , , , from each weakly adjacent (2 , , , whichcontradicts Eq. (3).Each 2-face ends with 2 + 2( ) = . Since G contains no 4 K and no T , , , the charge each face sendsacross each boundary edge is at most 2( ). Thus, when k ≥ k -face ends with at least k − k (2( )) = k ≥ . Since G contains no 3 C and no 3 C ◦ , each 4-face ends with at least 4 − ) = . It isstraightforward to check that each (1 , , , , − − = ,and each (2 , , − ) − ) = . It remains to check (2 , , , , xyz ends with less than . After (R1), face xyz has3 − ) = . Since xyz ends with less than , it receives at most by (R2) and (R3). So xyz must beadjacent to three 3-faces, and at most one of these is a (2 , , , , G contains no T ◦◦ , , , so the three 3-faces adjacent to xyz must share a new common vertex, say w .If one of wx, wy, wz is not contained in a string, then xyz is adjacent to two (2 , , ) by (R3), contradicting our assumption above. Thus we assume µ ( wx ) = µ ( wy ) = µ ( wz ) = 2.So G [ { x, y, z, w } ] contains a 2 K , contradicting Claim 1(a). This shows that each (2 , , , which completes the proof. / -flows: Proof of Theorem 1.4 In this section we prove Theorem 1.4. As in the previous section, this theorem is implied by the moretechnical result, Theorem 3.3. The proof of Theorem 3.3 is similar to that of Theorem 2.2, but with morereducible configurations and more details. 10 .1 Preliminaries on Modulo -orientations We define a weight function ρ as follows (which is similar to w in Definition 2.1). Definition 3.1.
Let P = { P , P , . . . , P t } be a partition of V ( G ). Let ρ G ( P ) = t X i =1 d ( P i ) − t + 31and ρ ( G ) = min { ρ G ( P ) : P is a partition of V ( G ) } . Analogous to Lemma 2.8, we have the following.
Lemma 3.2.
Let P = { P , P , . . . , P t } be a partition of V ( G ) with | P | > . Let H = G [ P ] and let Q = { Q , Q , . . . , Q s } be a partition of V ( H ) . Now Q ∪ ( P \ { P } ) is a refinement of P satisfying ρ G ( Q ∪ ( P \ { P } )) = ρ H ( Q ) + ρ G ( P ) − (31 − . (4) Proof.
The proof is identical to that of Lemma 2.8, with 17 in place of 11 and with 31 in place of 19.We typically assume that each edge has multiplicity at most 5 (since 6 K is strongly Z -connected,and so cannot appear in a minimal counterexample to Theorem 3.3, as we prove in Claim 9, below). Now ρ ( aK ) = 2 a − ρ ( T a,b,c ) = 2 a + 2 b + 2 c −
20, and ρ (3 K ) = −
1; see Figure 4. In each case, the minimumin the definition of ρ is achieved uniquely by the partition with each vertex in its own part. t t ... aaK t tt T a,b,c ba c t ttt K t ttt K +4 Figure 4:
The graphs aK , T a,b,c , K , K +4 . Let F = { aK : 2 ≤ a ≤ } ∪ { T a,b,c : 10 ≤ a + b + c ≤
11 and T a,b,c is 6-edge-connected . } It isstraightforward to check that no graph in F is strongly Z -connected. Further, if T a,b,c is 8-edge-connected,then k G k ≥ δ ( G ) / ≥
12. Thus, no graph in F is 8-edge-connected. The following theorem is the mainresult of Section 3. We call a partition P problematic problem-atic if G/ P ∈ F . Theorem 3.3.
Let G be a planar graph and β be a Z -boundary of G . If ρ ( G ) ≥ , then G admits a ( Z , β ) -orientation, unless G has a problematic partition. As easy corollaries of Theorem 3.3 we get the following two results. When a ≤
5, the graph aK has seven Z -boundaries and at most 6 orientations, so at least one boundary is not achievable.The graph 3 K cannot achieve the boundary β ( v ) = 0 for all v . In such an orientation D each vertex v must have | d + D ( v ) − d − D ( v ) | = 7. But now some two adjacent vertices must either both have indegree 8 or both have outdegree 8, and we cannot orientthe three edges between them to achieve this. For T a,b,c , it suffices to consider the case a + b + c = 11. Let V ( G ) = { v , v , v } .By symmetry, we assume d ( v ) ≤ d ( v ) ≤ d ( v ). For T , , , we cannot achieve β ( v ) = β ( v ) = 1 and β ( v ) = 5, since v and v must each have all incident edges oriented in. For T , , , we cannot achieve β ( v ) = 1, β ( v ) = 2, and β ( v ) = 4, since v must have all incident edges oriented in, and v must have all but one edges oriented in. For T , , , we cannot achieve β ( v ) = 1and β ( v ) = β ( v ) = 3, since v must have all incident edges oriented in. For T , , , we cannot achieve β ( v ) = β ( v ) = 2 and β ( v ) = 3, since v and v must each have all but one incident edge oriented in. heorem 3.4. Every -edge-connected planar graph is strongly Z -connected. Theorem 3.5.
Every odd- -edge-connected planar graph admits a modulo -orientation. In particular,every 16-edge-connected planar graph admits a modulo -orientation (and thus a circular 7/3-flow). The proofs of Theorems 3.4 and 3.5 are identical to those of Theorems 2.3 and 2.5, but with 17 in placeof 11 and with 31 in place of 19. Note that Theorem 3.5 includes Theorem 1.4 as a special case.For the proof of Theorem 3.3, we need the following two lemmas. Their proofs are more tedious thanenlightening, so we postpone them to the appendix. When a graph H is edge-transitive, we write H + or H − H + /H − to denote the graph formed by adding or removing a single copy of one edge. Lemma 3.6.
Each of the following graphs is strongly Z -connected: K , K +4 , and every 6-edge-connectedgraph T a,b,c where a + b + c = 12 . Let 5 C =4 5 C =4 denote the graph formed from 5 C by deleting a perfect matching. Lemma 3.7.
The graph C =4 is strongly Z -connected. Further, if G is a graph with | G | = 4 , k G k = 19 , µ ( G ) ≤ , and δ ( G ) ≥ , then G is strongly Z -connected. Let G be a counterexample to Theorem 3.3 that minimizes | G | + k G k . Thus Theorem 3.3 holds forall graphs smaller than G . This implies the following lemma, which we will use frequently. Lemma 3.8. If H is a planar graph with ρ ( H ) ≥ and | H | + k H k < | G | + k G k , then each of the followingholds.(a) If ρ H ( P ) ≥ for every nontrivial partition P , then H is strongly Z -connected unless H ∈ F .(b) If ρ ( H ) ≥ , then H is strongly Z -connected.(c) Assume that H is -edge-connected.(c-i) If ρ H ( P ) ≥ for every nontrivial partition P , then H is strongly Z -connected unless H ∼ = T a,b,c with a + b + c ∈ { , } .(c-ii) If ρ ( H ) ≥ , then H is strongly Z -connected.(c-iii) If H is -edge-connected, then H is strongly Z -connected.Proof. We apply Theorem 3.3 to H . (a) For each J ∈ F , the trivial partition Q ∗ satisfies ρ J ( Q ∗ ) ≤ max { − , − } = 7. Since ρ H ( P ) ≥ P , we knowthat H/ P / ∈ F . Part (b) follows immediately from (a). Consider (c). Since H is 6-edge-connected, theredoes not exist P such that | H/ P| = 2 and k H/ Pk ≤
5. For (c-i), suppose there is a nontrivial partition P such that H/ P ∼ = T a,b,c with a + b + c ∈ { , } . Now ρ H ( P ) = 2(11) − G is 8-edge-connected,so is G/ P , for each partition P . Recall that each element of F has edge-connectivity at most 7. Thus, G/ P / ∈ F .As in Section 2, the main idea of the proof is to show that ρ G ( P ) is relatively large for each nontrivialpartition P . This gives us the ability to apply Lemma 3.8 to subgraphs of G even after modifying themslightly, which yields more power when proving subgraphs are reducible. Claim 9. G has no strongly Z -connected subgraph H with | H | > . In particular,(a) G has no copy of K , K +4 , or a 6-edge-connected graph T a,b,c with a + b + c = 12 ; and(b) | G | ≥ . roof. The proof of the first statement is identical to that of Claim 1, with Z in place of Z . Note that (a)follows from the first statement and Lemma 3.6.Now we prove (b). Clearly | G | ≥
2, so first suppose | G | = 2. Since ρ ( G ) ≥
0, we know k G k ≥
2. Since G has no problematic partition, we know k G k ≥
6. But now G contains 6 K , which contradicts (a). So assume | G | = 3, that is G = T a,b,c . Since ρ ( G ) ≥
0, we know a + b + c ≥
10. Since G has no problematic partition, G is 6-edge-connected. By the definition of F , this implies that a + b + c ≥
12. Recall that G contains no6 K by (a); thus max { a, b, c } ≤
5. A short case analysis shows that G contains as a subgraph one of T , , , T , , , or T , , . Each of these has 12 edges and is 6-edge-connected, which contradicts (a). Claim 10. If P = { P , P , . . . , P t } is a nontrivial partition of V ( G ) , then(a) ρ G ( P ) ≥ ; and(b) ρ G ( P ) ≥ if P is normal.Proof. We argue by contradiction. For an almost trivial partition P , we have ρ G ( P ) ≥ ρ G ( V ( G )) − ≥
7, since G does not contain 6 K by Claim 9. If P = { V ( G ) } , then w G ( P ) = 0 −
17 + 31 = 14. Since | G | ≥ P = { P , P , . . . , P t } be a normal partition of V ( G ). We may assume | P | > H = G [ P ].For any partition Q = { Q , Q , . . . , Q s } of V ( H ), by Eq. (4) the refinement Q ∪ ( P \ { P } ) of P satisfies ρ H ( Q ) = ρ G ( Q ∪ ( P \ { P } )) − ρ G ( P ) + 14 . (5)(a) We first show that ρ G ( P ) ≥
7. If ρ G ( P ) ≤
6, then Eq. (5) implies that ρ H ( Q ) ≥ Q of H , since ρ G ( Q ∪ ( P \ { P } )) ≥
0. Hence ρ ( H ) ≥ H is strongly Z -connected by Lemma 3.8(b),which contradicts Claim 9. This proves (a).(b) We now show that ρ G ( P ) ≥
12. Suppose, to the contrary, that ρ G ( P ) ≤
11. If P contains at leasttwo nontrivial parts, say | P | >
1, then (a) implies ρ G ( Q ∪ ( P \ { P } )) ≥ Q of H . Hence ρ ( H ) ≥
10 by Eq. (5), and so H is strongly Z -connected by Lemma 3.8(b), which contradicts Claim 9.Assume instead that P contains a unique nontrivial part P and | P | ≥
3. For any nontrivial partition Q of H , the refinement Q ∪ ( P \ { P } ) of P is a nontrivial partition of G , and so ρ G ( Q ∪ ( P \ { P } )) ≥ ρ H ( Q ) ≥
10 for any nontrivial partition Q of H by Eq. (5). For the trivial partition Q ∗ of H ,since ρ G ( P ) ≤
11, Eq. (5) implies ρ H ( Q ∗ ) ≥
3. Since | H | = | P | ≥
3, we know H = aK . Since ρ ( H ) ≥ H = T a,b,c with a + b + c ≤
11. So Lemma 3.8(a) implies that H is strongly Z -connected, whichcontradicts Claim 9.The next two claims follow from Claim 10. They give lower bounds on the edge-connectivity of G . Claim 11.
For a partition P = { P , P , . . . , P t } ,(a) if | P | ≥ and | P | ≥ , then ρ ( P ) ≥ ; and(b) if | P | ≥ and | P | ≥ , then ρ ( P ) ≥ .Proof. Let H = G [ P ] and Q = { Q , Q , . . . , Q s } be a partition of H . By Eq. (4), ρ H ( Q ) = ρ G ( Q ∪ ( P \ { P } )) − ρ G ( P ) + 14 . (a) If ρ G ( P ) ≤
13, then ρ H ( Q ) ≥ Q of H since ρ G ( Q ∪ ( P \ { P } )) ≥ H is strongly Z -connected by Lemma 3.8(b), which contradicts Claim 9.(b) Similarly, if ρ G ( P ) ≤
18, then ρ H ( Q ) ≥ Q of H since ρ G ( Q ∪ ( P \ { P } )) ≥
12 byClaim 10(b). Again H is strongly Z -connected by Lemma 3.8(b), which contradicts Claim 9. Claim 12.
Let [ X, X c ] be an edge cut of G .(a) Now | [ X, X c ] | ≥ . That is, G is -edge-connected. b) If | X | ≥ and | X c | ≥ , then | [ X, X c ] | ≥ .Proof. (a) Let P = { X, X c } . Since | G | ≥ P is normal. Now Claim 10(b)gives 12 ≤ ρ G ( P ) = 2 | [ X, X c ] | −
34 + 31, which implies | [ X, X c ] | ≥ | X | ≥ | X c | ≥
3, then ρ G ( P ) ≥
19 by Claim 11(b). So 19 ≤ ρ G ( P ) = 2 | [ X, X c ] | −
34 + 31,which implies | [ X, X c ] | ≥ t tt t ttt t tt t t tt T , , T • , , T ◦ , , T , , Figure 5:
The graphs T , , , T • , , , T ◦ , , , T , , . Let T • , , T • , , denote the graph formed from T , , by subdividing an edge of multiplicity 1. We now showthat G contains none of the folllowing (shown in Figure 5) as subgraphs: T , , , T ◦ , , , T • , , , and T , , . Claim 13. G has no copy of T , , .Proof. Suppose G contains a copy of T , , with vertices x, y, z and µ ( xy ) = 5. We lift xz, zy to becomea new edge xy and contract the resulting 6 K induced by { x, y } . Let G ′ denote the resulting graph. Thetrivial partition Q ∗ of G ′ satisfies ρ G ′ ( Q ∗ ) ≥ ρ ( G ) − ≥
3. Every nontrivial partition Q ′ of G ′ corresponds to a normal partition Q of G in which the contracted vertex is replaced by { x, y } . Since xz, zy are the only two edges possibly counted in ρ G ( Q ) but not in ρ G ′ ( Q ′ ), we have ρ G ′ ( Q ′ ) ≥ ρ G ( Q ) − ≥ ρ ( G ′ ) ≥
3. Since G is 8-edge-connected by Claim 12, graph G ′ is 6-edge-connected, andso G ′ is strongly Z -connected by Lemma 3.8(c-ii). This is a lifting reduction of the first type. It shows that G is strongly Z -connected, which contradicts Lemma 9. Claim 14. | G | ≥ .Proof. Suppose the claim is false. Claim 9(b) implies | G | = 4. Since ρ ( G ) ≥
0, the trivial partition shows that k G k ≥
19. First suppose k G k >
19, and let G ′ = G − e , for some arbitrary edge e . Since k G ′ k < k G k , we willapply Lemma 3.8(c-i) to prove G ′ is strongly Z -connected. Since | G ′ | = 4, we know G ′ / ∈ F . So it sufficesto show that G ′ is 6-edge-connected and ρ G ′ ( P ) ≥ P . The first conditionholds because G is 8-edge-connected, by Claim 12(a). The second holds because ρ G ′ ( P ) ≥ ρ G ( P ) − ≥ G ′ is strongly Z -connected by Lemma 3.8(c-i), which contradicts Claim 9.Instead assume k G k = 19. Claim 12(a) implies δ ( G ) ≥
8. Since G contains no 6 K by Claim 9(a), weknow µ ( G ) ≤
5. Now Lemma 3.7 shows that G is strongly Z -connected. Thus, G is not a counterexample,which proves the claim. Claim 15. G has no copy of T ◦ , , .Proof. Suppose G contains a copy of T ◦ , , with vertices w, x, y, z and µ ( xy ) = 4. We lift xz, zy to become anew edge xy , and lift xw, wy to become another new edge xy , and then contract the resulting 6 K to forma new graph G ′ . The trivial partition Q ∗ of G ′ satisfies ρ G ′ ( Q ∗ ) ≥ ρ ( G ) − ≥
1. Every nontrivialpartition Q ′ of G ′ corresponds to a normal partition Q of G in which the contracted vertex is replaced by { x, y } . Since xz, zy, xw, wy are the only edges possibly counted in ρ G ( Q ) but not in ρ G ′ ( Q ′ ), Claim 10(b)implies ρ G ′ ( Q ′ ) ≥ ρ G ( Q ) − ≥
4. Since w = z , Claim 12(a,b) implies G ′ is 6-edge-connected. Because14 V ( G ′ ) | = | V ( G ) | − ≥
4, we know G ′ = T a,b,c with a + b + c ∈ { , } . Hence G ′ is strongly Z -connectedby Lemma 3.8(c-i). This is a lifting reduction of the first type. So G is strongly Z -connected, which is acontradiction. Claim 16. G has minimum degree at least . So G is -edge-connected by Claim 12.Proof. The second statement follows from the first. To prove the first, suppose there exists x ∈ V ( G ) with8 ≤ d ( x ) ≤
9. Let x , x be two neighbors of x . To form a graph G ′ from G , we lift x x, xx to becomea new edge x x , orient the remaining edges incident with x to achieve β ( x ), and finally delete x . This issimilar to achieving β ( v ) in the proof of Lemma 2.6 (that G has no copy of 6 K ). This is a lifting reductionof the second type. So, to show G has a β -orientation, it suffices to show that G ′ is strongly Z -connected.Observe that the trivial partition Q ∗ of G ′ satisfies ρ G ′ ( Q ∗ ) ≥ ρ ( G ) − −
1) + 17 ≥
1. Also, for analmost trivial partition Q ′ of G ′ with | Q | = 2, we have ρ G ′ ( Q ′ ) ≥ ρ G ′ ( Q ∗ ) + 17 − ≥
8. Note that when Q = { x , x } we still have µ G ′ ( x x ) ≤ Q ′ of G ′ , since Q = Q ′ ∪ { x } is a normal partition of G , we have ρ G ′ ( Q ′ ) ≥ ρ G ( Q ) − ≥
11. Since | G ′ | = | G | − ≥ ρ G ′ ( Q ′ ) ≥ G ′ is strongly Z -connected. Claim 17. G has no copy of T • , , .Proof. Suppose G has a copy of T • , , , with vertices v , v , v , v (in order around a 4-cycle) and µ ( v v ) = 5.We lift the edges v v , v v , v v to become a new copy of edge v v and contract the resulting 6 K ; callthis new graph G ′ . The trivial partition Q ∗ of G ′ satisfies ρ G ′ ( Q ∗ ) ≥ ρ ( G ) − ≥
1. Every nontrivialpartition Q ′ of G ′ corresponds to a normal partition Q of G in which the contracted vertex is replaced by { v , v } . Since v v , v v , v v are the only edges possibly counted in ρ G ( Q ) but not in ρ G ′ ( Q ′ ), we have ρ G ′ ( Q ′ ) ≥ ρ G ( Q ) − ≥ | G ′ | = | G | − ≥
4, so G ′ / ∈ F . Since G is 10-edge-connected by Claim 16, the graph G ′ is 6-edge-connected. So G ′ is strongly Z -connected byLemma 3.8(c-i). Claim 18. G has no copy of T , , .Proof. Suppose G contains a copy of T , , with vertices x, y, z and µ ( xy ) = 4. To form a new graph G ′ from G , we delete two copies (each) of xz, zy and add two new parallel edges xy , and then contract the resulting6 K induced by { x, y } . Claim 16 shows G ′ is 6-edge-connected. Similar to the proof of Claim 15, the trivialpartition Q ∗ of G ′ satisfies ρ G ′ ( Q ∗ ) ≥ ρ ( G ) − ≥
1, and every nontrivial partition Q ′ of G ′ satisfies ρ G ′ ( Q ′ ) ≥ ρ G ( Q ) − ≥
4. Since | G ′ | = | G | − ≥
4, Lemma 3.8(c-i) implies G ′ is strongly Z -connected.This is a lifting reduction of the first type, which implies that G is strongly Z -connected, and thus gives acontradiction. Claim 19.
For any normal partition P = { P , P , . . . , P t } with | P | ≥ , we have ρ G ( P ) ≥ . Proof.
Suppose the claim is false and let P be such a partition with ρ G ( P ) ≤
13. Let H = G [ P ]. Since G contains no copy of T , , or T , , , we know H = T a,b,c with a + b + c ∈ { , } (and min { a, b, c } ≥ | H | = | P | ≥
3, we know
H / ∈ F .Let Q = { Q , Q , . . . , Q s } be a partition of H . Now Q ∪ ( P \ { P } ) is a partition of G , and Eq. (4) implies ρ H ( Q ) = ρ G ( Q ∪ ( P \ { P } )) − ρ G ( P ) + 14 ≥ ρ G ( Q ∪ ( P \ { P } )) + 1. If Q is a nontrivial partition of H , then Q ∪ ( P \ { P } ) is a nontrivial partition of G , and so Claim 10(a) implies ρ H ( Q ) ≥ ρ G ( Q ∪ ( P \ { P } )) + 1 ≥ Q is the trivial partition of H , then ρ H ( Q ) ≥ ρ G ( Q ∪ ( P \ { P } )) + 1 ≥
1. By Lemma 3.8(a), the subgraph H is strongly Z -connected, which contradicts Claim 9.Now we can strengthen Claim 12(b). Claim 20. If [ X, X c ] is an edge cut with | X | ≥ and | X c | ≥ , then | [ X, X c ] | ≥ . tt ttt t tt tt t tt tt t (5 C =4 ) ◦◦ identified (5 C =4 ) ◦◦ T ◦◦◦ , , Figure 6: The graphs (5 C =4 ) ◦◦ , identified (5 C =4 ) ◦◦ , T ◦◦◦ , , . Proof.
Let X satisfy the hypotheses and let P = { X, X c } . We will prove ρ G ( P ) ≥
21. Assume, to thecontrary, that ρ G ( P ) ≤
20. Let H = G [ X ] and let Q = { Q , . . . , Q s } be a partition of H . Let P ′ = Q∪{ X c } .Eq. (4) implies ρ H ( Q ) = ρ G ( P ′ ) − ρ G ( P ) + 14. Since | X c | ≥
3, Claim 19 implies ρ G ( P ′ ) ≥
14. Thus ρ H ( Q ) ≥ −
20 + 14 = 8. By Lemma 3.8(b), subgraph H is strongly Z -connected, which contradictsClaim 10(b). So 21 ≤ ρ G ( P ) = 2 | [ X, X c ] | −
34 + 31, which implies | [ X, X c ] | ≥ G ′ is still 6-edge-connected. Thus, we will show that G ′ is strongly Z -connected, since it satisfies the hypotheses of Lemma 3.8(c-i).Recall that 5 C =4 5 C =4 denotes the graph formed from 5 C by removing the edges of a perfect matching. Claim 21. G contains neither a copy of (5 C =4 ) ◦◦ nor a copy of (5 C =4 ) ◦◦ with its two 2-vertices identified.Proof. Suppose G contains a copy of (5 C =4 ) ◦◦ with vertices v , v , v , v , w , w , where v , . . . , v lie on the4-cycle and N ( w ) = { v , v } and N ( w ) = { v , v } . In G we lift edges v w , w v to form a new copy of v v and lift edges v w , w v to form a new copy of v v ; call this new graph G ′ . In G ′ vertices v , . . . , v induce a copy of 5 C =4 (if either v v or v v is present in G , then G contains T ◦ , , , which is a contradiction).Claim 3.7 implies 5 C =4 is strongly Z -connected. Form G ′′ from G ′ by contracting { v , v , v , v } . Since G is 10-edge-connected by Claim 16, we know G ′′ is 6-edge-connected. The trivial partition Q ∗ of G ′′ satisfies ρ G ′′ ( Q ∗ ) ≥ ρ ( G ) + 3(17) − ≥
11. Each nontrivial partition Q ′′ of G ′′ corresponds to a normal partition Q of G in which the contracted vertex is replaced by { v , v , v , v } . Since at most four edges are countedin ρ G ( Q ) but not in ρ G ′′ ( Q ′′ ), we have ρ G ′′ ( Q ′′ ) ≥ ρ G ( Q ) − ≥ ρ ( G ′′ ) ≥
6, soLemma 3.8(c-ii) implies that G ′′ is strongly Z -connected, and also that G is strongly Z -connected, whichis a contradiction. If vertices w and w are identified, the same proof works, since Claim 16 still impliesthat G ′′ is 6-edge-connected. Claim 22. G contains no copy of T ◦◦◦ , , .Proof. Suppose G contains a copy of T ◦◦◦ , , with vertices v , v , v , w , w , w and d ( v i ) = 8 and d ( w i ) = 2for all i and N ( w i ) = { v , v , v } \ { v i } . Form G ′ from G by lifting the pair of edges incident to each vertex w i and contracting the resulting T , , . This is a lifting reduction of the first type. Since T , , is strongly Z -connected by Lemma 3.6, it suffices to show that G ′ is also strongly Z -connected. Claims 20 and 16imply that G ′ is 6-edge-connected. The trivial partition P ∗ of G ′ satisfies ρ G ′ ( P ∗ ) ≥ ρ ( G )+17(2) − ≥ P ′ of G ′ corresponds to a normal partition P of G in which the contracted vertex isreplaced by { v , v , v } . We show below that for such a partition we can strengthen Claim 19 to ρ G ( P ) ≥ ρ G ′ ( P ′ ) ≥ ρ G ( P ) − ≥ ρ G ( P )but not in ρ G ′ ( P ′ ). Thus, ρ ( G ′ ) ≥
3, so Lemma 3.8(c-ii) implies that G ′ is strongly Z -connected, which isa contradiction. Now it suffices to show that ρ G ( P ) ≥ ρ G ( P ) ≤
14. Let P be the part of P containing { v , v , v } , and let H = G [ P ]. We will show that H is strongly Z -connected, which gives a contradiction. Let Q = { Q , . . . , Q s } bea partition of H . Let P ′′ = Q ∪ ( P \ { P } ). Eq. (4) implies ρ H ( Q ) = ρ G ( P ′′ ) − ρ G ( P ) + 14 ≥ ρ G ( P ′′ ) ≥ Q is a nontrivial partition of H , then P ′′ is a nontrivial partition of G , so Claim 10 implies ρ H ( Q ) ≥ ρ G ( P ′′ ) ≥
7. Since H contains T , , by construction, and G does not contain T , , , we know that H / ∈ F . To apply Lemma 3.8(c-i), we show that H is 6-edge-connected. Consider a bipartition Q = { Q , Q } of H . Since Q is nontrivial, 7 ≤ ρ G ( P ′′ ) ≤ ρ H ( Q ) = 2 | [ Q , Q ] H | − | [ Q , Q ] H | ≥ H is 5-edge-connected. If H is 6-edge-connected, then Lemma 3.8(c-i) implies that H is strongly Z -connected, which is a contradiction. So assume H has a bipartition Q = { Q , Q } with | [ Q , Q ] H | = 5.By symmetry, we assume | Q | ≥ | Q | . Since H contains T , , and T , , is 6-edge-connected, we know that | Q | ≥
3. Now ρ G ( P ′′ ) = ρ G ( P ) + 2(5) − ≤ − P ′′ is normal with | Q | ≥
3, this contradictsClaim 10.
Fix a plane embedding of a planar graph G such that ρ ( G ) ≥
0. (We assume that all parallel edges betweentwo vertices v and w are embedded consecutively, in the cyclic orders, around both v and w .) If G hasa cut-vertex, then each block of is strongly Z -connected by minimality, so G is strongly Z -connected byLemma 1.6, which is a contradiction. Hence G is 2-connected. Since ρ ( G ) ≥
0, we have 2 k G k− | G | +31 ≥ | G | + | F ( G ) | − k G k = 2. Now solving for | G | and substituting into the inequality gives: X f ∈ F ( G ) ℓ ( f ) = 2 k G k ≤ | F ( G ) | − . We assign to each face f initial charge ℓ ( f ). So the total charge is strictly less than 34 | F ( G ) | /
15. To reach acontradiction, we redistribute charge so that each face ends with charge at least 34 /
15. We use the followingthree discharging rules.(R1) Each 2-face takes charge 2 /
15 from each weakly adjacent 3 + -face.(R2) Each 3-face takes charge 2 /
15 from each weakly adjacent 4 + -face with which its parallel edge hasmultiplicity at most 3 and 1 /
15 from each weakly adjacent 4 + -face with which its parallel edge hasmultiplicity 4.(R3) After (R1) and (R2), each 3-face with more than 34 /
15 splits its excess equally among weakly adjacent3-faces with less than 34 / /
15. By (R1) each 2-face ends with 2 + 2(2 /
15) =34 /
15. Consider a 5 + -face f . Since G contains no copy of 6 K , each edge of f has mutliplicity at most 5.Since G contains no copy of T , , , face f sends at most 4(2 /
15) across each of its edges. Thus f ends withat least ℓ ( f ) − / ℓ ( f ) = 7 ℓ ( f ) / ≥ /
15. Consider a 4-face f . Since G contains no copy of T • , , ,each edge of f has multiplicity at most 4. So f sends at most 3(2 /
15) + 1 /
15 = 7 /
15 across each of its edges.If f sends at most 5/15 across one edge, then f ends with at least 4 − / − /
15 = 34 /
15. If f sendsat most 6 /
15 across at least two of its edges, then f ends with at least 4 − / − /
15) = 34 /
15. Soassume that neither of these cases holds. Thus, each edge of f has multiplicity 4, and f is weakly adjacentto 3-faces across at least three of its edges. This contradicts Claim 21.Let f be a 3-face T a,b,c . If a + b + c ≤
8, then f ends (R2) with at least 3 − (8 − /
15) = 35 / a + b + c ≥
9. Since G has no T , , , we know max { a, b, c } ≤
4. Since G has no T , , , ifmax { a, b, c } = 4, then min { a, b, c } = 1. Thus, each 3-face T a,b,c finishes (R1) with excess charge at least1 /
15 unless T a,b,c ∈ { T , , , T , , } . So we only need to consider T , , and T , , . Suppose f is T , , . Eachface adjacent to f across an edge of multiplicity 4 is not a 3-face, since G has no T ◦ , , . So f ends (R2) withat least 3 − (9 − /
15) + 2(1 /
15) = 35 /
15. Hence, each 3-face f ends (R2) with at least 35 /
15 unless f is T , , . 17o assume that f is T , , . If any adjacent face is not a 3-face, then f ends (R2) with at least 3 − (9 − /
15) + 2 /
15 = 35 /
15. So assume each adjacent face is a 3-face. If these three adjacent faces do notintersect outside f , then G contains a copy of T ◦◦◦ , , , a contradiction. If all three faces intersect outside f , then | V ( G ) | = 4, which contradicts Claim 14. So assume that exactly two faces adjacent to f intersectoutside f . Let f and f denote the 3-faces adjacent to f that intersect outside f . Denote the boundariesof f , f , and f by (respectively) vwx , vwy , and wxy . Suppose µ ( wy ) = 3. Now f and f each end (R2)with at least 35 /
13, so by (R3) each gives f at least (1 / / f ends happy. So assume µ ( wy ) = 3.Now d ( w ) = 3 + 3 + 3, which contradicts that δ ( G ) ≥
10, by Claim 16. This completes the proof.
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Each of the following graphs is strongly Z -connected: K , K +4 , and every 6-edge-connectedgraph T a,b,c where a + b + c = 12 .Proof. Throughout we fix a Z -boundary β and construct an orientation to achieve β .Let G = 6 K , with V ( G ) = { v , v } . To achieve β ( v ) ∈ { , , , , , , } , the number of edges we orientout of v is (respectively) 3, 0, 4, 1, 5, 2, 6.Let G = T a,b,c , with a + b + c = 12 and δ ( G ) ≥
6. (We handle this before 3 K +4 .) Let V ( G ) = { v , v , v } . If G contains a 6-vertex, say v , then µ ( v v ) = 6. Since G/v v ∼ = 6 K is strongly Z -connected, G is strongly Z -connected by Lemma 1.6(ii). So assume that δ ( G ) ≥
7. If G contains a 7-vertex v i and β ( v i ) = 0, then weorient 5 edges incident to v i to achieve β ( v i ), and lift the remaining pair of nonparallel edges to form a newedge. We are done, since 6 K is strongly Z -connected. If G contains an 8-vertex v j and β ( v j ) / ∈ { , } , thenwe orient 4 edges incident to v j to achieve β ( v j ), and lift two pairs of nonparallel edges to form new edges.Again we are done, since 6 K is strongly Z -connected. Since k G k = 12 and δ ( G ) ≥
7, the possible degreesequences of G are (a) { , , } , (b) { , , } , and (c) { , , } . The edge multiplicities of G are the threevalues k G k − d ( v i ). So G is (a) T , , , (b) T , , , or (c) T , , . In each case we assume d ( v ) ≤ d ( v ) ≤ d ( v ).In (a) we may assume β ( v ) = β ( v ) = 0, which implies β ( v ) = 0. To achieve this boundary, orient alledges out of v and all edges into v . In (b) we may assume β ( v ) = 0, β ( v ) = 1, and β ( v ) = 6. To achievethis boundary, orient all edges out of v and all edge into v . (If instead β ( v ) = 6 and β ( v ) = 1, then wereverse the direction of all edges.) In (c) we assume β ( v i ) ∈ { , } for all i . This yields a contradiction, since P i =1 β ( v i ) ≡ G = 3 K +4 , with V ( G ) = { v , v , v , v } and d ( v ) = d ( v ) = 9 and d ( v ) = d ( v ) = 10. Similar tothe previous paragraph, we may assume β ( v ) = β ( v ) = 0, β ( v ) = 1, and β ( v ) = 6. (If not, then wecan lift some edges pairs at v i and use the remaining edges incident to v i to achieve β ( v i ).) To achieve thisboundary, start by orienting all edges out of v , all edges into v , and all edges v v out of v . Now reverseone copy of v v and reverse one copy of v v . Lemma 3.7.
The graph C =4 is strongly Z -connected. Further, if G is a graph with | G | = 4 , k G k = 19 , µ ( G ) ≤ , and δ ( G ) ≥ , then G is strongly Z -connected.Proof. Assume G satisfies the hypotheses (either the first or second), and let V ( G ) = { v , v , v , v } . Ourplan is to form a new graph G i from G by lifting one, two, or three pairs of edges incident to v i , using theremaining edges incident to v i to achieve the desired boundary β ( v i ) at v i . This is a lifting reduction ofthe second type. If k G i k ≥
12 and G i is 6-edge-connected, then G i is strongly Z -connected by Lemma 3.6,and so we can find an orientation to achieve the β boundary of G . We will show that in every case we canconstruct such a G i , and achieve β ( v i ) using edges incident to v i that are not lifted to form G i .Denote V (5 C =4 ) by { v , v , v , v } , with N ( v ) = N ( v ) = { v , v } , and fix a Z -boundary β . If β ( v ) ∈{ , , , } , then we lift three pairs of edges incident to v and use the remaining edges to achieve β ( v ).Notice that the resulting graph G satisfies k G k = 12, and we are done in this case. So, by symmetry,we assume β ( v i ) ∈ { , , } for each i . The possible multisets of β values are { , , , } , { , , , } , and { , , , } . Up to symmetry, we have five possible Z -boundaries. Figure 7 shows orientations that achievethese.
000 0 <<<> <<<<< <>>> <<<<<
520 0 <<>> <<<<< <>>> <<<<<
020 5 <<<> <<<<> >>>> <<<<<
225 5 <<>> <<<<< <<>> >>>>>
525 2 <<>> <<<<< <<>> <<<<<
Figure 7: Orientations achieving the possible boundaries with β ( v i ) ∈ { , , } for all i .
01 6 < <<<< >>>> < < < < <>>> >> Case 1 000 0 >>>> <> <<< >>>> <<> < < < Case 2 010 6 <> <<<< >>>> < < < >>>> <> Case 3 001 6 <> <<< >>>> < < < < >>> <>> Case 4Figure 8: In each case v is at top, v center, v left, and v right. Now we prove the second statement. Suppose G contains an 8-vertex v i . To form G i , we lift one (arbitrary,nonparallel) pair of edges incident to v i . Now k G i k = 19 − G i contains a copy of 6 K , then weare done by Lemma 1.6, since 6 K is strongly Z -connected, and contracting this copy of 6 K yields another6 K . So instead we assume µ ( G i ) ≤
5. The edge-connectivity of G i is δ ( G i ) = k G i k − µ ( G i ) ≥ − G i is 6-edge-connected, we are done by Lemma 3.6. Hence, we assume that δ ( G ) ≥ v i , v j of vertices has no edges joining it; that is, µ ( v i v j ) = 0. By symmetry, we assume i = 1 and j = 2. Since d ( v ) ≥ d ( v ) ≥
9, we get that µ ( v v ) + µ ( v v ) ≥ µ ( v v ) + µ ( v v ) ≥ G has no 6 K , each edge of the 4-cycle v v v v has multiplicity at least 4. Either µ ( v v ) = 5 or µ ( v v ) = 5; by symmetry we assume the latter. If µ ( v v ) = 1, then we lift edge v v , v v to form anew copy of v v . We contract the resulting 6 K induced by { v , v } . The resulting graph G ′ is T , , , sowe are done by Lemmas 1.6 and 3.6. Instead assume µ ( v v ) = 0. Now G = 5 C − (formed from 5 C bydeleting a single edge). Thus G contains 5 C =4 as a spanning subgraph, and so G is strongly Z -connectedby Lemma 3.6. Thus, we assume µ ( v i v j ) ≥ i, j ∈ [4].Suppose µ ( v i v j ) = 5 for some distinct i, j ∈ [4]; by symmetry, say µ ( v v ) = 5. Since µ ( v v ) ≥ µ ( v v ) ≥
1, we lift one copy of each of v v and v v to form a new copy of v v , and then contract { v , v } (calling the new vertex w ). Denote this new graph by G ′ . We show that G ′ is strongly Z -connected, whichimplies the result for G by Lemma 1.6, since 6 K is strongly Z -connected. We first show that G is 8-edge-connected. Each edge cut separating a single vertex v i has size d ( v i ) ≥ δ ( G ) ≥
8. If an edge cut S separates G into two parts of size 2, then | S | ≥ k G k − µ ( G ) ≥ − G is 8-edge-connected, whichimplies that G ′ is 6-edge-connected. Since k G k = 19, we have k G ′ k = 19 − G ′ is strongly Z -connected, by Lemma 3.6. Thus G is strongly Z -connected by Lemma 1.6(ii). This implies that µ ( v i v j ) ≤ i, j ∈ [4].Suppose that µ ( v i v j ) = 1 for some pair i, j ∈ [4]; say µ ( v v ) = 1. Since d ( v ) ≥ d ( v ) ≥ µ ( G ) ≤
4, we have µ ( v v ) = µ ( v v ) = µ ( v v ) = µ ( v v ) = 4. Since k G k = 19, this implies µ ( v v ) = 2;see Case 1 in Figure 8. By orienting 5 edges incident to a vertex v i we can achieve any boundary value β ( v i ) other than 0. So if β ( v ) = 0 or β ( v ) = 0, then we achieve it by orienting 5 incident edges, andlifting two pairs of incident edges to reduce to a 6-edge-connected subgraph G i with k G i k = 12. Similarly,by orienting 4 edges incident to a vertex v i we can achieve any boundary value at v i other than 1 or 6. So if β ( v ) / ∈ { , } or β ( v ) / ∈ { , } , then we achieve β ( v i ) by orienting 4 edges incident to v i and lifting 3 pairsof incident edges; we do this so that the three newly created edges in G i are not all parallel. Since µ ( G ) ≤ µ ( G i ) ≤
6. Now we can finish on G i , by Lemma 3.6. Thus, by symmetry between v and v , weassume β ( v ) = β ( v ) = 0, β ( v ) = 1, and β ( v ) = 6. Case 1 in Figure 8 shows an orientation achieving thisboundary. So in what remains we assume that µ ( v i v j ) ≥ i, j ∈ [4].Since k G k = 19 and δ ( G ) ≥
9, the degree sequence is either { , , , } or { , , , } . Suppose weare in the first case. By symmetry, we assume d ( v ) = 11, µ ( v v ) = µ ( v v ) = 4, and µ ( v v ) = 3. Since d ( v ) = d ( v ) = d ( v ) = 9 and µ ( v v ) + µ ( v v ) + µ ( v v ) = 8, we have µ ( v v ) = 2 and µ ( v v ) = µ ( v v ) = 3. See Case 2 of Figure 8. If β ( v i ) = 0 for any i ∈ { , , } , then we achieve β ( v i ) by orienting 5edges incident to v i , and we lift two pairs of incident edges to form G i , which is 6-edge-connected and has k G i k = 12. So we assume β ( v ) = β ( v ) = β ( v ) = 0. This implies that also β ( v ) = 0. Case 2 in Figure 8shows an orientation achieving this boundary. 20inally, assume the degree sequence is { , , , } and µ ( v i v j ) ≥ i, j ∈ [4]. If µ ( v i v j ) ≥ i, j ∈ [4] then G ∼ = 3 K +4 , which contradicts Lemma 3.6. So assume by symmetry that µ ( v v ) =2. First suppose that d ( v ) = 10. This implies µ ( v v ) = µ ( v v ) = 4. Since each edge has multiplicity2, 3, or 4, we cannot have d ( v ) = 10 (because otherwise µ ( v v ) = 1). So d ( v ) = 9 and, by symmetrybetween v and v , we assume d ( v ) = 9 and d ( v ) = 10. This implies that µ ( v v ) = 3, µ ( v v ) = 4, and µ ( v v ) = 3; see Case 3 of Figure 8. As above, we can lift two or three pairs of incident edges if either β ( v ) = 0, β ( v ) = 0, β ( v ) / ∈ { , } , or β ( v ) / ∈ { , } . So we assume β ( v ) = β ( v ) = 0, β ( v ) = 1, and β ( v ) = 6. (If, instead, β ( v ) = β ( v ) = 0, β ( v ) = 6, and β ( v ) = 1, then we can achieve this by reversingevery edge.) The desired orientation is shown in Case 3 of Figure 8.Again assume the degree sequence is { , , , } and that µ ( v v ) = 2. Rather than as above, we nowassume d ( v ) = d ( v ) = 9. So d ( v ) = d ( v ) = 10. By symmetry between v and v (and also between v and v ) we assume µ ( v v ) = µ ( v v ) = 3, µ ( v v ) = µ ( v v ) = 4, and µ ( v v ) = 3. For the same reasonsas in the previous paragraph, we assume β ( v ) = β ( v ) = 0, β ( v ) = 1, and β ( v4