Flows in Graphs: Tutte s Flow Conjectures

Transcription

1 Master s Thesis Rikke Marie Langhede Flows in Graphs: Tutte s Flow Conjectures Thesis for the Master degree in Mathematics. Department of Mathematical Sciences, University of Copenhagen. Speciale for cand.scient graden i matematik. Institut for matematiske fag, Københavns Universitet. Supervisor: Bergfinnur Durhuus. June nd, 017

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3 Abstract In this thesis we study flows in graphs. Among the major open problems in modern graph theory are the 5-Flow Conjecture, which states that every bridgeless graph admits a 5-flow, and the 3-Flow Conjecture which states that every 4-edge-connected graph admits a 3-flow. This thesis reviews the most important results in flow theory, including comprehensive proofs and state-of-the-art techniques in the field. We will introduce the standard results in flow theory in order to prove some partial results of the flow conjectures above, for example the 6-Flow Theorem and the reduction of the 3-Flow Conjecture to 5-edge-connected graphs. A particularly important result is the Weak 3-Flow Theorem which was recently proved by Thomassen with bound k = 8 on the edge-connectivity. It has afterwards been improved by Lovász et al. with a bound of k = 6. We prove the theorem with the smaller of the bounds. Finally, we will consider generalizations of the flow concept to group connectivity, circular flows, and flow with values in a prescribed set, where we will see the extensive consequences the proof of the Weak 3-Flow Theorem has on the field of flow theory. Resume I dette speciale kigger vi på flows i grafer. Blandt de største uløste problemer i moderne grafteori findes 5-Flows-Formodningen, som siger at enhver broløs graf har et 5-flow, samt 3-Flows-Formodningen, som siger at at enhver 4-kantssammenhængende graf har et 3-flow. Dette speciale gennemgår de vigtigste resultater i flow-teori, herunder dybdegående beviser og de nyeste teknikker i feltet. Vi introducerer klassiske resultater i flow-teori med henblik på at bevise delresultater for formodningerne ovenfor, for eksempel 6-Flows-Sætningen og reduceringen af 3-Flows-Formodningen til 5-kantssammenhængende grafer. Et særligt vigtigt resultat er den Svage 3-Flows-Sætning, som for nyligt er blevet bevist af Thomassen med en nedre grænse på kant-sammenhæng på k = 8. Det er efterfølgende blevet forbedret til k = 6 af Lovász et al. Vi beviser sætningen for k = 6. Til sidst vil vi kigge på generaliseringer af flows til gruppesammenhæng, cirkulære flows og flows med værdier i en bestemt mængde, hvor vi vil se de vidtrækkende konsekvenser beviset for den Svage 3-Flows-Sætning har for hele flow-teorien. 1

4 Contents 1 Introduction Notation and preliminary definitions Preliminary results Menger s Theorem Nash-Williams - Tutte s Theorem Mader s Lifting Theorem Flows 13.1 Orientations Flow-colouring duality The flow polynomial k-flows and Z k -flows Some properties of flows Sums of flows Products of flows Flows on modified graphs k-flows for small k Example: The flow numbers of the complete graph Tutte s flow conjectures Equivalent expressions and minimal counterexamples Progress on the 5-Flow conjecture The 8-Flow Theorem The 6-Flow Theorem Reduction of the 3-Flow Conjecture The Weak 3-Flow Conjecture A result dependent on the number of odd vertices The Weak 3-Flow Theorem Thomassen s strategy The strategy by Lovász et al The proof of the Weak 3-Flow Theorem Consequences The sharpness of the result Group connectivity Properties of group connectivity Inheritance of flow properties Equivalence of group connectivity with a fixed order Monotonicity Z -connectivity Z 3 -connectivity A-connectivity for A = A-connectivity for A

5 6 Circular flows The Circular Flow Conjectures Circular colourings Modulo k-orientations and circular ( + 1/k)-flows Flows with values in a prescribed set 85 8 Appendix 89 3

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7 1 Introduction A flow in a graph is defined by giving each edge an orientation and a value, such that the sum of the values around each vertex under the orientation is zero. The concept was introduced by Tutte [4], [5] as a generalization of colouring problems, and the big question of the topic is how many different values we need to do this. The 5-Flow and 3-Flow Conjectures of Tutte asserts that it can be done with only the values {1,, 3, 4} for all bridgeless graphs and the values {1, } for 4-edge-connected graphs, respectively. These conjectures are among the major open problems in modern graph theory and they appear in many standard textbooks, such as [], [4], [7], [6], and [11]. In this thesis we introduce the standard results in flow theory, we prove some partial results of the flow conjectures above, and we consider certain generalizations of the flow concept. In particular, we prove the following conjecture: There is a natural number k such that every k-edge-connected graph admits a 3-flow. This conjecture was proved for k = 6 in the article by Lovász et al., [18], and the proof of this is of such general nature that it has far-reaching consequences in many other directions than the one initially intended. When considering the generalizations of the flow concept, we will see this. The topic of flows is a field of research which is constantly changing. While rapidly producing new results, this also means that no standard textbook is up to date on the subject. There exists a few surveys on the topic, for example two by Jaeger; [7] from 1979 and [9] from 1988, but both of these are way too old to contain the newest proofs and techniques. There exists another survey written by Lai, [15], which is from 015, but this is entirely without any explanations or proofs. This thesis fills a gap in this regard. It is a complete review of the most important results, it includes comprehensive proofs and state-of-the-art techniques, and it uses the same notation throughout the paper. It is the hope of the author that this thesis may provide an easily understandable introduction to flows and related concepts to all readers new to the subject. New results in this thesis: In [17] Lai et al. studies 3-flows in graphs of high edge-connectivity dependent on the number of odd vertices. They show that k-edge-connected graphs with t > 0 odd vertices admit a 3-flows if k 4 log t. Lovász et al. later showed in [18] that all 6-edge-connected graphs admit 3-flows. Comparing this to the work of Lai et al., we see that graphs which has odd vertices admit 3-flows if they are 4-edge-connected, and graphs which have 4 or more odd vertices admit 3-flows if they are 6-edge-connectivity. We improve these results for graphs which have or 4 odd vertices, showing that only -edge-connectivity is needed in the first case and only 5-edge-connectivity is needed in the second case. To the best of our knowledge, these results are the best known for graphs with or 4 odd vertices. The thesis is structured as follows: In section 1, we define the basic notation and state some theorems known from other areas of graph theory. In section, we will define flows and the basic properties of flows. We consider a connection 5

8 between flows and colourings in plane graphs in order to verify the importance of flows. Then we prove two theorems which will imply that admitting an integer flow of a certain size is equivalent to admitting group flows when the groups are of the same size. We prove some properties of flows, including some properties concerning sums and products of flows and flows on modified graphs. Eventually, we consider k-flows for small k s and find the flow numbers of the complete graphs. In section 3, we will introduce Tutte s flow conjectures and a generalization to modulo k-orientations, known as the Modulo k-orientation Conjecture. Then we prove some propositions which states that we may show the 5-Flow Conjecture and 3-Flow Conjecture for graphs with certain less general properties. This is done by considering properties of a minimal counterexample. We consider the progress on the 5-Flow Conjecture, proving first that all graphs admit 8-flows and then that all graphs admit 6-flows. Finally, we reduce Tutte s 3-flow conjecture to a similar statement about 5-edge-connected graphs. In section 4, we consider the Weak 3-Flow Conjecture. We first prove a version of this which is dependent on the number of odd vertices in graph. This result requires a large edge-connectivity and we prove that for certain small numbers of odd vertices it can be done better. Then we consider the work by Thomassen, [], and Lovász et al., [18]. Both prove the weak 3-flow conjecture but with different bounds on the edge-connectivity and a slight difference in the statements of their main theorems which we show are equivalent. In fact, they prove a generalization of the Weak 3-Flow Conjecture to general modulo k-orientations. The proofs use a special technique where some claims about the smallest counterexamples are proved such that in the end we can conclude that no smallest counterexample can exist. We only include the proof by Lovász et al., [18], as it has the lowest bound on edge-connectivity. In section 5, we introduce a generalization of flows which is already known from the proof of the Weak 3-Flow Theorem, namely group connectivity. We show some basic properties of group connectivity, and then we consider which of the properties of flows that are still valid for group-connectivity. In particular, we will see what consequences the proof of Lovász et al., [18], has for Z 3 -connectivity of 6-edge-connected graphs. In section 6, we introduce circular flows, which is another generalization of flows that restricts the number of possible values on the edges. The conjectures of Tutte translate to conjectures about circular flows, and we consider in what extent we can prove these. In this section the proof of Lovász et al., [18], has further consequences, namely that any 6k-edgeconnected graph admits a circular ( + 1/k)-flow. In section 7, we prove a result which considers the existence of a flow in a graph when the possible edge-values are restricted to a set which is not an entire group as earlier assumed. This result is very different from any other result in the thesis and might introduce a new way of thinking of flows. 1.1 Notation and preliminary definitions In this thesis, G denotes a graph which may have parallel edges and loops. G is undirected unless else is specified. We let V (G) denote the set of vertices of G and E(G) denote the set of edges of G. Note, that we might use the notation e = uv to indicate that the edge e connects the vertices u and v. One may also use the notation uv to represent an edge, but this is ambiguous as several edges may connect the same pair of vertices in a graph which allows parallel edges. Hence, we will only use it to specify the end-vertices of an edge. 6

9 As we will need the definition of an orientation of a graph throughout the rest of the thesis, we will define it formally here. First, we define a digraph. Definition 1.1. A directed graph D (also called a digraph) is an ordered pair (V (D), E(D)) consisting of disjoint sets, a set V (D) of vertices and a set E(D) of edges, together with an incidence function ψ D : E(D) V (D) that associates with each edge of D an ordered pair of vertices. If ψ D (e) = (u, v) for some edge e E(D) and vertices u, v V (D), then e connects u and v and is said to be directed towards v or, equivalently, e is directed away from u. We also say that u is the tail of e while v is its head, or that e is an ingoing edge at v and an outgoing edge at u. Definition 1.. Given a graph G, one may obtain a digraph D from G by replacing each edge in G by just one of the two possible directed edges with the same end-vertices. We say that D is an orientation of G. Note, that given an orientation D of a graph G, we will still refer to the resulting graph as G, so when we talk about the orientation of an edge in G we refer to the orientation of the corresponding edge in D. We will denote by E(x) the set of edges incident to a vertex x in a graph G. Given an orientation of G, E + (x) (resp. E (x)) denotes the set of edges directed away x (resp. the set of edges directed towards x). This definition can be extended to vertex sets X V (G), such that E + (X) (resp. E (X)) denotes the set of edges with their tails in X and heads not in X (resp. the set of edges with their heads in X and tails not in X) when G is given an orientation. In general, for vertex sets X, Y V (G), we denote by E(X, Y ) the set of edges having exactly one end in X and the other in Y, while E + (X, Y ) (resp. E (X, Y )) denotes the edges which has their tails in X and heads in Y (resp. the edges with their heads in X and tails in Y ) when G is given an orientation. We will denote by d(x) the degree of x, and likewise, given an orientation of G, d + (x) (resp. d (x)) is defined as the outgoing (resp. ingoing) degree of x. For vertex subsets X V (G) we define d(x) = E(X, V (G) \ X). Let G be a graph. If V (G) > k and G X is connected for every vertex set X V (G) with X < k, then we say that G is k-connected. Likewise, if V (G) > 1 and G \ F is connected for every edge set F E(G) with F < k, then we say that G is k-edge-connected graph. If (V 1, V ) is a partition of V (G), then the set E(V 1, V ) of edges crossing this partition is called a cut. A cut of size k is called a k-cut, and a 1-cut is called a bridge. Note, that a graph G is k-edge-connected if and only if it has no l-cuts for l < k. For other standard terminology and notation we refer to [] or [4]. 1. Preliminary results The theory we will explore in the thesis is build on the foundation of understanding connectivity and the existence of certain paths in a graph. For that purpose, the following three theorems are essential. 7

10 1..1 Menger s Theorem The following theorem is one of the most fundamental theorems in graph theory. The proof can be found in the appendix. Note, that two or more paths are said to be independent if none of them contains an inner vertex of another. Theorem 1.3 (Menger). Let G be a graph. Then: (i) G is k-connected if and only if it contains k independent paths between any two vertices. (ii) G is k-edge-connected if and only it contains k edge-disjoint paths between any two vertices. 1.. Nash-Williams - Tutte s Theorem Another result which will be used many times in this thesis is a corollary of the following theorem discovered simultaneous by Nash-Williams and Tutte in 1961, see [0]: Theorem 1.4 (Nash-Williams - Tutte). A graph G contains k edge-disjoint spanning trees if and only if for every partition P of the vertex set V (G) it has at least k( P 1) cross-edges. Note, that given a partition of the vertex set of a graph, we define the cross-edges of this partition to be the edges whose ends lie in different partition sets. The proof below follows the strategy of [4, Section 3.5]. Proof. Let G be a graph and let some k N be given. One implication is easy: Suppose G contains k edge-disjoint spanning trees and let P be a partition of V (G). As every spanning tree contains at least P 1 cross-edges (which is the case as they also span G/P ), G has at least k( P 1) cross-edges in total. For the other implication we will start with some definitions. Let F be the set of all k-tuples F = (F 1,..., F k ) of edge-disjoint spanning forests in G with the maximum total number of edges, i.e. such that E(F ) := E(F 1 )... E(F k ) is as large as possible. Next, we describe what it means for a k-tuple F to be obtained from the replacement of some edge by another: Note first, that if F = (F 1,..., F k ) F and e E(G) \ E(F ), then F i + e contains a cycle for every i = 1,..., k. Otherwise, if F i does not contain a cycle we can replace F i by F i + e in F to obtain a k-tuple of edge-disjoint spanning forests F with E(F ) > E(F ), a contradiction. So we let e E(G) \ E(F ) and consider an edge e e in the cycle in F i + e for some fixed i. Putting F i := F i + e e and F j = F j for j i, we obtain another k-tuple F := (F 1,..., F k ) which is in F - we say that F has been obtained from F by the replacement of the edge e by e. Note, that for every x y-path in F i there is a unique x y-path in F i. We now consider a fixed k-tuple F 0 = (F1 0,..., F k 0) F. We denote by F 0 the set of all k-tuples which can be obtained from F 0 by a series of edge replacements. Let also E 0 := F F 0 (E(G) \ E(F )) 8

11 x u e' ufv i v xf'y i y Figure 1: The paths xf i y and uf iv in C 0. i.e. E 0 is the set of edges which is not in any of the edge-disjoint spanning forests of F for some F F 0. Let G 0 = (V (G), E 0 ). Claim 1. Let F = (F 1,..., F k ) F 0 and let F = (F 1,..., F k ) be obtained from F by the replacement of an edge of F i. If x, y are the ends of a path in F i C0, where C 0 is a component of G 0, then the path from x to y in F i is in C 0 as well. Let e = uv be the (only) edge in E(F i ) \ E(F i). We assume that e is on the x y-path in F i (denoted by xf i y), since otherwise xf iy = xf i y and there would be nothing to show. It suffices to show that uf i v C 0, since then (xf i y e ) uf i v is a connected subgraph of F i C 0 that contains x and y and hence also xf i y. Let e be any edge in uf i v. Then e E 0 since we can replace e in F F 0 by e and obtain an element of F 0 not containing e. Thus, uf i v G 0, and hence uf i v C 0 since u, v xf i y C0 (Figure 1 shows how it may look). This proves Claim 1. Claim. For every e 0 E(G) \ E(F 0 ) there exists a vertex set X V (G) which is connected in F 0 i for every i = 1,..., k and contains the ends of e 0. As F 0 F 0, we have e 0 E 0. Let C 0 be the component of G 0 containing e 0. We will prove that the vertex set X := V (C 0 ) has required properties. Let i {1,..., k} be given. We show that for every edge xy C 0, the path xfi 0y exists and lies in C 0. As C 0 is connected, the union of all these paths is a connected spanning subgraph of Fi 0 [X], implying that X is connected in F 0 i. So let e C 0 be given, where e connects vertices x and y. As e E 0, the definition of E 0 implies there exists an N N and k-tuples F n = (F1 n,..., F k n ) for n = 1,..., N such that each F n is obtained from F n 1 by an edge replacement and F N satisfies e E \ E(F N ). We now use Claim 1 on F N and any k-tuple F obtained from F N by an replacement of any possible edge in Fi N with the edge e: Since F i contain a path from x to y in C0 (namely e), the path from x to y in Fi N is in C 0 as well. Applying Claim 1 to the pairs F n and F n 1 for n = N,..., 1 yields xfi 0y C0 as wanted. We now prove the theorem by induction on V (G). For V (G) = the assertion obviously holds. So suppose that for every partition P of V there are at least k( P 1) cross-edges. We will construct k edge-disjoint spanning trees in G. 9

12 Pick a k-tuple F 0 = (F1 0,..., F k 0 ) F. If every F 0 i is a tree, we are done. If not, we have E(F 0 ) = k i=1 E(F 0 i ) < k( V (G) 1) since one of the Fi 0 s must be a forest but not a tree and thus has E(F 0 i ) < V (G) 1. On the other hand, we have E(G) k( V (G) 1) as the assumption holds for the partition of V (G) into single vertices. Thus, there exists an edge e 0 E(G) \ E(F 0 ). By Claim, there exists a vertex set X V (G) which is connected in every Fi 0 and contains the ends of e 0 - in particular, X. The two graphs X and G/X are considered in turn. Since every partition of the contracted graph G/X induces a partition of G with the same cross-edges, G/X also has at least k( P 1) cross-edges with respect to any partition P, so by the induction hypothesis, G/X therefore has k edge-disjoint spanning trees T 1,..., T k. In each spanning tree T i of G/X we now replace the vertex v X contracted from X with the spanning tree Fi 0 G[X] (which are indeed spanning trees as Fi 0 G[X] are connected by Claim ), and thus, we obtain k edge-disjoint spanning trees in G. It is not too clear why this theorem is so useful, but in the corollary below it becomes obvious as k-edge-connectedness for some k N is a very reasonable requirement for a graph. In fact, this requirement appears in Tutte s Flow Conjectures as we shall see later. Corollary 1.5. Every k-edge-connected graph G has k edge-disjoint spanning trees. Proof. If G is a k-edge-connected graph, every partition set in a vertex partition of G is joined to other partition sets by at least k edges. Hence, for any partition into r sets, G has at least 1 r i=1 k = kr cross-edges, and thus G has k edge-disjoint spanning trees by Theorem Mader s Lifting Theorem A third result concerns the edge-connectivity of a graph after a so-called lifting. Let us start by defining a lifting: Let z be a vertex of a graph G. If e 1 = zx and e = zy are two edges and x and y are distinct vertices, then the deletion of the edges e 1 and e and the addition of the edge e = xy is called the lifting of e 1 and e at z (see Figure ). x x z e e 1 z e y y Figure : The lifting of e 1 and e at z. The following is a result by Mader, proved in We will not include the proof here, but refer instead to [19]. 10

13 Theorem 1.6. Suppose that z is a non-separating vertex of a graph G which has degree 4 and at least two different neighbours. Then we can lift a pair of edges from z such that for any two distinct vertices x, y not equal to z, the maximal number of edge-disjoint paths between x and y is not decreased. In the corollary below we see the consequences this theorem has for the edge-connectivity in a graph. Corollary 1.7. If G is k-edge-connected, and z is a non-separating vertex which has degree at least k + and at least two different neighbours, then we can lift a pair of edges from z such that the resulting graph is still k-edge-connected. If z is a non-separating vertex which has even degree and at least d(z)/ different neighbours, then we can lift all the edges incident with z such that the resulting graph minus z is k-edge-connected. Proof. Suppose we lift a pair of edges from the vertex z where d(z) k + and let G be the resulting graph. Let C be a minimal edge-cut in G and let U 1 and U be the (nonempty) components of G C. We can assume that z U 1. Assume for contradiction that C < k. For every pair of vertices x, y V (G ) \ {z} there are k edge-disjoint paths from x to y by Theorem 1.6, so x and y must be contained in the same component. Therefore, either U 1 = {z} and U = V (G) \ {z}, or U 1 = V (G) and U =. Hence U 1 = {z} and U = V (G) \ {z}. But this implies that d(z) = C < k, a contradiction. We conclude that G is k-edge-connected. The proof of the other statement is as follows. For d(z) = it is easy to check as the lifting at z corresponds to a contraction of one of the edges incident to z. As contractions preserves edge-connectivity, the resulting graph minus z is k-edge-connected. For d(z) 4 we repeatedly lift a pair of edges from z and use Theorem 1.6, preserving the maximal number of edge-disjoint paths between any pair of vertices x and y distinct from z, until d(z) = where we do as described above. Thus, when we delete z there are k edge-disjoint paths between any pair of vertices in the resulting graph. This implies that the resulting graph is k-edge-connected. 11

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15 Flows We define the notion of a flow, originally introduced by Tutte in [4], [5]. Definition.1. Let G be a graph, and let A be an Abelian group. We define an A-flow (D, f) in G as an orientation D of G together with an edge-function f : E A \ {0} which satisfies Kirchhoff s law f(e) f(e) = 0 (1) on all vertices x V (G). e E + (x) e E (x) In other literature one might see a distinction between a flow and a nowhere-zero flow (we say that a flow is nowhere-zero if the edge-function f satisfies f(x) 0 for all x E(G)), but in this thesis all flows are nowhere-zero and we will instead use the notion A-circulations for A-flows which are not necessarily non-zero. Given a graph G, we define the support of an edge-function f : E(G) A to be the set of edges in G where f is non-zero, i.e. supp(f) = {e E(G) f(e) 0}. Hence, an A-flow (D, f) is an A-circulation which has supp(f) = E(G). When the group A is the integers we may say an integer flow instead of a Z-flow (similarly, we may say a integer circulation instead of a Z-circulation). For an example of a flow in a graph, see Figure Figure 3: An integer flow in K 4. Note, that given an orientation of the graph, we will use the notation f + (x) = f(e) f (x) = e E + (x) e E (x) for x V (G). Hence we can express Kirchhoff s law for short as f + (v) f (v) = 0 for all v v(g). Likewise, for vertex sets X V (G), we define f + (X) = f(e) f (X) = e E + (X) e E (X) f(e) f(e) It turns out there is a very similar relationship between the two definitions above. 13

16 Proposition.. If (D, f) is a circulation, then f + (X) f (X) = 0 for every vertex subset X V. Proof. It obviously holds that f + (X) = x X e E + (x) f(e) e E(X,X) f(e) and f (X) = x X e E (x) f(e) e E(X,X) f(e), and since f satisfies Kirchhoff s law in D, we must have f + (X) f (X) = f(e) f(e) x X e E + (x) e E(X,X) f(e) f(e) x X e E (x) e E(X,X) = f(e) f(e) = 0. x X Thus, f + (X) f (X) = 0 as wanted. e E + (x) e E (x) As the statement f + (X) f (X) = 0 holds for every vertex subset X V (G), it clearly implies f + (x) f (x) = 0 for all x V (G), as well, so we see that the two statements are equivalent. k-flows It is of great interest to not just study integer flows with arbitrary edge-functions, but instead consider integer flows whose edge-functions are bounded. This motivates the definition of a k-flow: Definition.3. Let k be a natural number. A nowhere-zero integer flow (D, f) whose flow function f satisfies 0 < f(e) < k for all e E(G) is called a k-flow. The following property of k-flows is a direct consequence of this definition: Proposition.4. If a graph admits a k-flow then it also admits a (k + l)-flow for any l 0. Thus, we can define the flow number of a graph: Definition.5. Given a graph G, the least integer k such that G admits a k-flow is called the flow number of G and is denoted by φ(g). If G does not admit a k-flow for any k 1, we define φ(g) =. Loops Assume some graph G admits a flow and that G contains a loop e incident to a vertex x. Given an orientation of G, e is clearly both in the ingoing and outgoing edge set of x, and hence the net flow of x is independent of the flow value of e. Therefore, any loop in G might be deleted without any consequences for the flow on G. Unless otherwise mentioned, we will thus assume all graphs in this thesis to be loopless and any loop arising from a contraction of some edge will automatically be deleted. 14

17 .1 Orientations In this subsection we will consider the importance of the orientation of the edges. It turns out that the existence of a flow depends on the underlying structure of the graph, not the orientation. Note, that when we need to emphasize in which orientation D a certain edge-function f satisfies Kirchhoff s law (assuming that such one exists), we will write f D instead of f. Lemma.6. Let (D, f) be an A-flow of a graph G where A is an Abelian group. If D is the orientation obtained from D by reversing the orientation of the edge e 0 and f is the edge-function given by f (e 0 ) := f(e 0 ) and f (e) := f(e) otherwise, then (D, f ) is another A-flow on G. Proof. This is obvious, as f is non-zero and f D (e) f D (e) = e E + (x) holds for all vertices x V (G). e E (x) e E + (x) f D (e) e E (x) f D (e) = 0 Note, that this lemma still holds if A-flow is replaced by A-circulation. Now, using the above lemma an appropriate number of times we get the following proposition: Proposition.7. Let A be an Abelian group. There exists an A-flow in G if and only if there for any orientation D of G exists an edge-function f such that (D, f) is an A-flow in G. This means that it is the underlying structure of a graph that determines whether or not a graph admits an A-flow and not the specific orientation of the edges of the graph. Thus, flows are a property of undirected graphs. The proposition justifies the following phrase: We will say that an edge-function f belongs to an A-flow in G if there exists an orientation D of G such that (D, f) is an A-flow in G. We say that an integer flow (D, f) is positive if the edge-function f is positive. A particularly nice corollary of Lemma.6 is the following. Corollary.8. A graph admits an integer flow if and only if it admits a positive integer flow. Proof. Given any integer flow (D, f), we let D be the orientation obtained from D by reversing all edges e such that f(e) < 0. Then the edge-function f obtained as in Lemma.6 is clearly positive, hence (D, f ) is a positive flow.. Flow-colouring duality In this subsection we will explore a connection between colourings and flows in plane graphs. First we will define the dual graph. We will denote the set of faces of a plane graph G by F (G). Definition.9. Let G be a plane graph. We define the dual graph G by letting each vertex f in G correspond to a face f in G and letting each edge e in G correspond to an edge e in G. Two vertices f 1 and f are joined by the edge e in G if and only if the corresponding faces f and g lie on each side of the edge e in G. 15

18 One can show that the dual of a plane graph is a plane graph itself, and that each vertex v in G correspond to a face v in G No further introduction to plane duality will be given here, but can be found in [4, Section 4.6]. One should note, though, that any cycle in a plane graph translates to an edge-cut in the dual graph (and vice versa) (see [4, Prop ]). For good measure, we will define what it means for a graph to be k-colourable and k-facecolourable. Definition.10. Let G be a graph, and let S be a set of colours. A vertex-colouring (or just colouring) of G is a map c : V (G) S which assigns to each vertex v V (G) a colour c(v) from S. The colouring c is said to be proper if any pair u, v of adjacent vertices are assigned different colours, i.e. c(u) c(v). G is said to be k-colourable if there is a proper colouring of the vertices of G requiring at most k colours, hence we can assume S = {1,,..., k}. Definition.11. Given a graph G, the least integer k such that G has a proper k-colouring is called the chromatic number of G and is denoted by χ(g). If G does not have a k-colouring for any k 1, we define χ(g) =. Definition.1. Let G be a plane graph, and let S be a set of colours. A face-colouring of G is a map c : F (G) S which assigns to each face f F (G) a colour c(f) from S. The colouring c is said to be proper if for each edge e E(G) the faces f, g on either side of e are assigned different colours, i.e. c(f) c(g). G is said to be k-face-colourable if there is a proper face colouring of G requiring at most k colours, hence we can assume S = {1,,..., k}. We will now prove two theorems connecting the k-flows and the k-face-colourings of a graph. Both follow the strategies of the proofs in [7, Section 1.4] Note, that when we refer to the left and the right side of an edge with orientation, we mean the side of the edge which is to the left or the right when the edge has the head directed upwards (see Figure 4). Right Left Right Left Figure 4: The left and right side of an edge. Theorem.13. Let G be a plane graph. If G is k-face-colourable, then G admits a k-flow. Proof. Let c : F (G) {1,..., k} be a proper face-colouring of G. Let D be the orientation of G which orients each edge e E(G) such that the face with the smaller colour is on the right side of e. Define the edge-function f : E(G) {1,..., k 1} such that f(e) = 16

19 c(f 1 ) c(f ) for each edge e E(G), where F 1, F are the two faces incident with e. Hence, f(e) = c(f 1 ) c(f ) when F 1 is on the left side of e and F is on the right side, and f(e) = c(f ) c(f 1 ) otherwise. We will show that (D, f) is a positive k-flow. Clearly, 0 < f(e) < k for each edge e E(G) as c is a proper face-colouring. Hence, we only need to show that f satisfies Kirchhoff s law in D. For a vertex v V (G), denote the edges in E(v) by e 1,..., e t in a clockwise order around v, and let F i be the face with e i and e i+1 consecutively on its boundary for i = 1,..., t (for simplicity we define e t+1 := e 1 and F t+1 := F 1 ). See Figure e 5 e t F 4 F t e 4 v e1 F F 3 1 F e e 3 Figure 5: The vertices and faces around v. Let ε i = { 1 if e i has its tail at v, 1 if e i has its head at v. Thus, ε i = 1 implies that F i 1 is on the left side of e i and F i is on the right, and ε i = 1 implies it is the other way around for i =,..., t + 1. Then f(e i ) = ε i (c(f i 1 ) c(f i )), i.e. for i =,..., t + 1. Hence for i = 1,..., t, which for i = t becomes This implies that By the definition of ε i, we get that e E + (v) c(f i ) = c(f i 1 ) ε i f(e i ) c(f i ) = c(f t ) c(f t ) = c(f t ) f(e) i ε j f(e j ) j=1 t ε i f(e i ). i=1 t ε i f(e i ) = 0. i=1 e E (v) f(e) = t ε i f(e i ) = 0. i=1 and therefore Kirchhoff s law is satisfied at v as wanted. 17

20 Theorem.14. Let G be a bridgeless plane graph. If G admits a k-flow then G is k-facecolourable. Proof. Let D be an arbitrary orientation of G, and let (D, f) be a k-flow in G. We will define a face colouring c : F (G) {0,..., k 1} as follows. Start at an arbitrary face F 0 and let c(f 0 ) = 0. For each edge e E(G), we let F and F be the two faces incident with e where F is on the left side of e and F is on the right. If only one of F and F is coloured already, then we put c(f ) c(f ) + f(e) (mod k). We immediately see, that the colouring is proper if it is also well-defined. Thus, we only need to show that no face can receive multiple different colours during the process depending on which neighbouring face we colour from. So suppose that at some point during the colouring of the faces F 1 is a uncoloured face incident with two faces F and F 3 which are already coloured. Let e be the edge on the boundary of F 1 and F, and let e 3 be the edge on the boundary of F 1 and F 3. We can assume without loss of generality that F 1 is on the left side of the edges e and e 3 (as the method from Proposition.7 prescribes that we can merely change the sign of the value of the edge-function when reversing an edge). We need to show that c(f ) + f(e ) c(f 3 ) + f(e 3 ) (mod k). () In the dual graph G of G, the vertex subset R = {F V (G ) F F (G), F is coloured already} induces a subgraph of G which must be connected since we only colour faces adjacent to other coloured faces. Thus, there is a path in R from F1 to F, which implies there is a cycle T in G containing F1, F and F 3 and the edges e and e 3 which has V (T ) \ {F1 } R. Since the dual of the cycle T is an edge-cut T in G, let T separate V (G) into the two nonempty vertex sets X and V (G) \ X. Since (D, f) is a flow, we get from Proposition. that f + (X) f (X) = 0, i.e. e E + (X) f(e) e E (X) f(e) = 0. (See how it may look on Figure 6.) F F e 3 e 1 F 3... T... X Figure 6: The faces F 1, F and F 3 and the edge-cut T. Now, as the equation c(f i ) c(f i ) + f(e i) (mod k) holds for all neighbouring pairs of faces which shares an edge in T, we get by a telescope sum on the edges of T that c(f ) + f(e ) c(f 3 ) + f(e 3 ) + f(e) f(e) (mod k) e E + (X) e E (X) 18

21 which implies eq. () above. Thus, the colouring is well-defined, and so G is k-face-colourable. Theorem.15. For every dual pair G, G of plane graphs, χ(g) = φ(g ) Proof. This follows from Theorem.13 and Theorem.14 when we note that a k-facecolouring of a graph G is a k-colouring of the dual graph G. Note, that instead of requiring the graphs to be embeddable in the plane, one might require them to be embeddable in the sphere or other orientable surfaces (under certain assumptions) to obtain similar results. This is beyond the scope of this thesis, though, and we refer to [7] for more information on this topic..3 The flow polynomial In this section loops are allowed (as we shall see, they are essential for the following proof). By the following theorem, the number of flows on a given graph depends on a polynomial which we will call the flow polynomial. The proof below follows the strategy of that in [4, Section 6.3]. Theorem.16. Let G be a graph. There exists a polynomial P such that, for any finite Abelian group A and any orientation D of G, the number of A-flows on G with the orientation D is P ( A 1). Proof. We use induction on the number of edges in G which are not loops. Assume first all edges in G are loops. Then there are A 1 possible values at each edge, since Kirchhoff s law is always satisfied no matter which value we assign a loop and since the value should be nonzero. Thus, if we put m := E(G), the number of A-flows on G is ( A 1) m which implies that the polynomial P (t) = t m is the one in question. Now, assume that there exists an edge e 0 = xy which is not a loop, and let D be any orientation of G. Let F denote the set of A-flows on G with orientation D, let F 1 be the set of A-circulations with orientation D on G which are zero on e 0 and non-zero on all other edges, and let F be the set of A-circulations with orientation D on G which are non-zero expect possibly on e 0. Obviously, F is the disjoint union of F and F 1, so F = F F 1. By the induction hypothesis, there are polynomials P G e0 and P G/e0 such that the number of A-flows on G/e 0 and G e 0 with orientation D (where D is the restriction of D to E(G) \ {e 0 }) is P G e0 (k) and P G/e0 (k), respectively, where k := A 1. So if we can show that F 1 = P G e0 (k) and then we are done as we get F = P G/e0 (k), F = P G/e0 (k) P G e0 (k) 19

22 hence the number of A-flows on G with a given orientation D is P G (k), where P G := P G/e0 P G e0 is clearly a polynomial. So we want to prove that F 1 corresponds exactly to the A-flows on G e 0 and F corresponds exactly to the A-flows on G/e 0. The first assertion is easy: When we have a A-circulation on G which is zero on e 0 and non-zero on all other edges, then we get an A-flow on G e 0 (with orientation D ), since Kirchhoff s law still holds at all vertices, and vice versa. The second statement requires a little more work. Consider an A-circulation (D, f) in F. Let v 0 denote the vertex which is the contraction of the vertices x and y in G/e 0, and let f be the restriction of f to G/e 0. Then (D, f ) is indeed an A-flow on G/e 0 as Proposition. with X := {x, y} imply that Kirchhoff s law holds at v 0. Meanwhile, given an A-flow (D, f ) in G/e 0, defining an A-circulation (D, f) in F requires the definition of f(e 0 ) as we just put f(e) = f (e) for e E(G) \ {e 0 }. Kirchhoff s law is then satisfied at all vertices apart from x and y, so if we can define f(e 0 ) cleverly we are done. The direction of e 0 is already determined by D, so assume without loss of generality that it is directed towards y. Put f(e 0 ) := f(e) f(e) e E + (x)\{e 0 } e E (x) to ensure that Kirchhoff s law holds at x. Using Kirchhoff s law at v 0 in G/e 0, we get that 0 = f + (v 0 ) f (v 0 ) = (f + (x) + f + (y) f(e 0 )) (f (x) + f (y) f(e 0 )) = f + (y) f (y). Thus, Kirchhoff s law holds at y as well, and F corresponds exactly to the flows on G/e 0 as wanted. As the proof above do not use any of the structure from A other than the order, the following corollary is obvious. Corollary.17. If A and A are two finite Abelian groups of same order, then the graph G admits an A-flow if and only if it admits an A -flow. This is a very interesting result! One would think that the number of flows in a graph varies with the structure of the chosen group, but apparently this is not so..4 k-flows and Z k -flows One very important tool when determining whether a certain graph admits a k-flow for some k is the following connection between k-flows and Z k -flows: Theorem.18. A graph admits a k-flow if and only if it admits a Z k -flow. The proof follows the strategy in [, Section 1.]. Proof. Let G be a graph. Obviously, a k-flow (D, f) is also a Z k -flow, since if the flow function f satisfies f + (v) f (v) = 0 in D for some v V (G) it also implies f + (v) f (v) 0 0

23 (mod k) in D when the values of f are considered as elements in Z k by the natural mapping. Thus, we only need to prove the other implication. Assume G has a Z k -flow (D, f). If we regard the elements of Z k as elements of Z, the flow function f has the following properties: f(e) {1,..., k 1} for all e E, f + (v) f (v) 0 (mod k) for all v V. Define a vertex v to be positive, balanced or negative if the net flow f + (v) f (v) is positive, zero or negative, respectively. Since ( f + (v) f (v) ) = f(e) f(e) v V (G) = e E(G) f(e) e E(G) v V (G) e E + (v) f(e) = 0, e E (v) either all vertices are balanced or there are both positive and negative vertices. Define the excess of f to be v V f + (v) f (v). Then f is a k-flow if and only if the excess of f is 0. If this is the case, we are done. Otherwise, suppose the excess of f is positive, i.e. not all vertices are balanced. We will show how we from (D, f) can obtain a new Z k -flow (D, f ) such that f has a strictly smaller excess than f. Consider a positive vertex x V (G) and let X V (G) be the set of all vertices which can be reached from x through a directed path in D, x included. Then E + (X) =, so ( f + (v) f (v) ) = f(e) f(e) v X = e E + (X) f(e) v X e E (X) e E + (v) f(e) = e E (X) e E (v) f(e) 0 as f is positive. This implies that there exists a negative vertex in X, say y. By the construction of X, there is a directed path P from x to y in D. Let D be the orientation obtained from D by reversing the direction of all edges of P and keeping the direction of all other edges, and let f be the edge-function obtained from f by setting f (e) := k f(e) for all edges e in P and f := f otherwise. Then (D, f ) is also a Z k -flow on G, but the excess of f is k less than the excess of f. Iterating this process results in a Z k -flow which has excess 0, hence a k-flow in G. This completes the proof. Combining Theorem.18, Corollary.17 and Corollary.8 yields the following powerful theorem: Theorem.19. Let G be a graph and let k be a positive integer. The following properties are equivalent: (i) G admits a k-flow. (ii) G admits a positive k-flow. (iii) G admits an A-flow where A is a finite Abelian group of order k. 1

24 This theorem is the single most important tool for determining whether a graph admits a certain flow, as it is often easier to determine whether a graph admits a Z k -flow compared to determining whether a graph admits a k-flow. We shall see multiple uses of the theorem in the next subsection..5 Some properties of flows Having established the most basic properties of flows, we shall continue on to some of the easier properties. Proposition.0. If a graph G has a bridge, then it admits no flow. Proof. Assume for contradiction that G has an A-flow (D, f) for some Abelian group A, and let e be a bridge in G. As f is nowhere-zero, we have f(e) 0. Let X be one of the two components of G e. Then f + (X) f (X) 0 as it reduces to either f(e) or f(e), which contradicts Proposition.. Hence, (D, f) cannot be an A-flow in G. Proposition.1. Let G be a graph and let A, A be Abelian groups. If (D, f) is an A- circulation on G and g : A A is a group homomorphism, then (D, g f) is an A -circulation in G. Proof. We only need to check that Kirchhof s law is satisfied. We have: (g f)(e) (g f)(e) = g f(e) f(e) = g(0) = 0 e E + (v) e E (v) e E + (v) for all v V (G), and thus (D, g f) is an A -circulation in G..5.1 Sums of flows e E (v) Proposition.. Let G be a graph, and let A be an Abelian group. The sum of two A-circulations on G which has the same orientation is another A-circulation. Proof. Let (D, f 1 ) and (D, f ) be A-circulations with the same orientation D of G. Defining f := f 1 + f, Kirchhoff s law is still satisfied at all vertices: Let v V (G), then f + (v) f (v) = (f + 1 (v) + f + (v)) (f 1 (v) + f (v)) = (f + 1 (v) f 1 (v)) + (f + (v) f (v)) = 0 in D as (D, f 1 ) and (D, f ) satisfy Kirchhoff s law. Thus, (D, f) is an A-circulation. This proposition proves to be even more useful when we can prove that the sum is nowherezero. Under this condition the proposition extends to prove that the sum of two A-circulations is a A-flow. One version of this is the following: Corollary.3. Let G be a graph, and let A be an Abelian group. Suppose G = G 1 G where E(G 1 ) E(G ) = and G 1 admits the A-flow (D 1, f 1 ) and G admits the A-flow (D, f ). If we define the edge-function f to be equal to f 1 on G 1 and f on G, then (D 1 D, f) is an A-flow in G.

25 We can now prove the opposite statement of Proposition.0, that each bridgeless graph have a flow number which is not infinity. Corollary.4. Let G be a bridgeless graph. Then G admits a k-flow for some k 1. Proof. As G is bridgeless, all edges e E(G) are contained in some cycle e C e in G. If D is any orientation on G, there clearly exists -circulations (D, f e ) which assigns the values ±1 to the edges of C e and 0 to all other edges in G. Let E(G) := {e 1, e,..., e m } where m := E(G). Then f := m i=1 i 1 f ei must be nowhere-zero, and thus Proposition. implies that (D, f) is a k-flow on G, where k = E(G)..5. Products of flows Another interesting property is products of flows: Proposition.5. Let G be a graph, let D be any orientation, and let k 1, k be integers. Then G admits a k 1 k -flow if and only if G admits a k 1 -circulation (D, f 1 ) and a k -circulation (D, f ) such that supp(f 1 ) supp(f ) = E(G). Proof. For the first implication, note that Theorem.19 implies that if G admits a k 1 - circulation and a k -circulation then G admits a Z k1 -circulation (D, f 1 ) and a Z k -circulation (D, f ). Thus, we easily see that (D, (f 1, f )) is a Z k1 Z k -flow in G as it is nowhere-zero and Kirchhoff s law clearly holds. For the other implication, suppose G admits a k 1 k -flow. Then G admits a Z k1 Z k -flow (D, f). Let f 1 : E(G) Z k1 and f : E(G) Z k be edge-functions such that f(e) = (f 1 (e), f (e)). As f is nowhere-zero, we must have supp(f 1 ) supp(f ) = E(G). Furthermore, f 1 and f clearly satisfy Kirchhoff s law at all vertices, hence (D, f 1 ) is a Z k1 -circulation on G and (D, f ) is a Z k -circulation on G. Since (D, f i ) is a Z ki -flow on the graph induced by supp(f i ) for i = 1,, Theorem.18 implies that G admits a k 1 -circulation (D, f 1 ) with supp(f 1 ) = supp(f 1) and a k -circulation (D, f ) with supp(f ) = supp(f ), hence supp(f 1 ) supp(f ) = E(G), as wanted. In fact, one can easily prove the following statement: Proposition.6. If (D, f 1 ) is a k 1 -circulation and (D, f ) is a k -circulation on a graph G and supp(f 1 ) supp(f ) = E(G), then (D, k f 1 + f ) and (D, k 1 f + f 1 ) are k 1 k -flows in G. An equivalent way of stating Proposition.5 is as following: Corollary.7. Let G be a graph and let k 1, k be integers. Then G admits a k 1 k -flow if and only if G = G 1 G, where G i admits a k i -flow for i = 1,..5.3 Flows on modified graphs Proposition.8. Let G be a graph, and let e E(G). If G admits an A-flow for some Abelian group A, then G/e admits an A-flow as well. Proof. Let (D, f) be an A-flow on G, and let e = xy. Consider the restriction f of f to E(G/e). We will prove that (D, f ) is an A-flow on G/e, where D is the restriction of D to G/e. 3

26 Clearly, f is an edge-function which takes values in A \ {0} and it satisfies Kirchhoff s law on all vertices except possibly on the contraction of e, v 0. But as Proposition. holds in G on the set {x, y}, Kirchhoff s law holds on v 0 as well, so (D, f ) is indeed a A-flow in G/e. Proposition.9. Let G be a bridgeless graph and let e 0 be an edge of G. If G e 0 admits a k-flow, then G admits a (k + 1)-flow. Proof. Let (D, f) be a positive k-flow on G e 0. Let e 0 = xy, and denote by X the set of vertices (x included) in G e 0 which can be reached by from x through a directed path in D. Assume y / X. Then E + (X) =, and as G is bridgeless there is a path from y to x in G e 0, which implies that X cannot be a component in G e 0. Hence f + (X) f (X) = f (X) < 0, (3) as f is strictly positive. This contradicts Proposition., so y X. Hence, we have a positive directed path P from x to y. We now obtain a (k + 1)-flow (D, f ) on G by defining D as the extension of D to E(G), giving e 0 the direction from y to x. We define f by f(e) for e E(G) \ (P {e 0 }), f (e) = f(e) + 1 for e P, 1 for e = e 0. Then f takes values in {1,..., k}, and Kirchhoff s law is clearly satisfied at all vertices as f is the sum of f and an integer circulation which is 1 on the (directed) cycle P {e 0 } and zero otherwise. Hence (D, f ) is indeed a (k + 1)-flow on G. The proof above also shows that it there is a directed path in D between any pair of vertices. As all graphs admit a k-flow for some k as shown in Corollary.4, and any k-flow induces a positive k-flow by Corollary.8, we get the following statement: Corollary.30. Any bridgeless graph has an orientation D which is a strongly connected digraph. We will prove another statement, inspired by [9, Section 8], which is very similar to Proposition.9. But first, we need a lemma. Lemma.31. Let G be a bridgeless graph and let e 0 E(G). Let k {, 3, 4}. If G admits a k-flow, then it admits a positive k-flow (D, f) such that f(e 0 ) = 1. Proof. If k = the statement is obvious. If k = 3, let (D, f) be a positive 3-flow in G. Suppose f(e 0 ) 1. Let C be a directed cycle in G which contains e 0 (such one exists by the proof of Proposition.9). Define a new edge-function f : E(G) {1, } by f (e) := { 3 f(e) for e C, f(e) otherwise. 4