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Circular flow number of generalized Blanuša snarks
Discrete Mathematics 313 (2013) 975–981
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Circular flow number of generalized Blanuša snarks Robert Lukot’ka Department of Mathematics and Computer Science, Trnava University in Trnava, Trnava, 918 43, Slovakia
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Article history: Received 17 May 2011 Received in revised form 9 January 2013 Accepted 14 January 2013 Available online 8 February 2013 Keywords: Circular flow number Real flow number Generalized Blanuša snarks
abstract A circular flow on a graph is an assignment of directions and flow values from R to the edges so that for each vertex the sum of the flow values on exiting edges equals the sum of the flow values on entering edges. A circular nowhere-zero r-flow is a circular flow φ with flow values satisfying 1 ≤ |φ(e)| ≤ r − 1 for each edge e. The circular flow number of a graph G is the infimum of all reals r such that G has a circular nowhere-zero r-flow. We prove that the circular flow number of all generalized Blanuša snarks except for the Petersen graph is 4.5. We also bound the circular flow number of Goldberg snarks, both from above and from below. © 2013 Elsevier B.V. All rights reserved.
1. Introduction and preliminaries A k-flow is an assignment of directions and integer flow values to the edges so that the flow conservation condition is satisfied. The flow conservation condition asserts that for each vertex the sum of the flow values on exiting edges equals the sum of the flow values on entering edges. A nowhere-zero k-flow is a k-flow that does not use value 0. The concept of nowhere-zero flows can be extended to the real numbers. This extension can be done in multiple different ways; nevertheless, all the approaches suggested by various authors [1,3,9] turned out to be equivalent. Perhaps the most natural way to introduce the real numbers to nowhere-zero flows is to directly employ real numbers instead of integers as the flow values. Let r ≥ 2 be a real number. A circular nowhere-zero r-flow φ on a graph G is an assignment of directions and real flow values from [1; r − 1] to the edges of G so that the flow conservation condition holds. The circular flow number of a graph G (also known as the real flow number [5,6]) is the infimum of all reals r such that G has a circular nowhere-zero r-flow. This infimum is attained and it is rational [1]. Moreover, the ordinary flow number is the ceiling of the circular flow number [1]. The exact value of the circular flow number is only known for few classes of graphs. This includes e.g. eulerian, regular bipartite, and complete graphs [9]. If we concentrate on cubic graphs, the situation however is much more clearer, at least for 3-edge-colourable graphs. Every bipartite 3-edge-colourable graph has nowhere-zero 3-flow. According to Steffen’s gap theorem [9], every cubic graph with circular flow number strictly less than 4 is bipartite. Since 3-edge-colourable graphs have a nowhere-zero 4-flow, a non-bipartite 3-edge-colourable graph has circular flow number 4. However, only very little is known about the circular flow number of non-3-edge-colourable cubic graphs. A bridgeless non-3-edge-colourable cubic graph is called a snark. Snarks are an intensively studied class of graphs, probably due to the fact that for many important conjectures in graph theory, any minimal counterexample must be a snark. One of these conjectures is Tutte’s 5-flow Conjecture—every bridgeless graph has a nowhere-zero 5-flow [10]. Indeed, Kochol [4] showed that any minimal counterexample to Tutte’s 5-flow Conjecture must be a cyclically 6-edge-connected snark of girth at least 11. Recent results answer several questions about the circular flow number of snarks [5–7]. However, these result use very specific graphs and tell us very little about how to tackle the problem of determining the circular flow number of a given
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Fig. 1. A nowhere-zero (4 + 1/k)-flow on the Goldberg snark Hk .
snark. To develop new methods we chose well-known infinite families of snarks with a simple global structure. Perhaps the best example of such snarks are the Isaacs snarks, whose circular flow number was determined in [5]. The main objective of this article is to determine the circular flow number of generalized Blanuša snarks, which are two families of snarks with a similar ‘‘circular’’ structure. As the main result of this paper, we show that the circular flow number of all generalized Blanuša snarks except for the Petersen graph is 4.5. Moreover, we bound the circular flow number of graphs from another well-known class of snarks, the Goldberg snarks. Goldberg snarks are constructed like the well-known Isaacs snarks. Instead of a Y -shaped block we use the block depicted in Fig. 1 bottom. We found a nowhere-zero (4 + 1/k)-flow on the Goldberg snark G2k+1 , similar to a nowhere-zero (4 + 1/k)flow on the Isaacs snark I2k+1 (Fig. 1). Together with a natural lower bound for the circular flow number of a snark [5] we know that the circular flow number of the Goldberg snark G2k+1 is in the interval [4 + 1/(2k + 1); 4 + 1/k].
2. Generalized Blanuša snarks The generalized Blanuša snarks of type 1 and type 2 are two important classes of snarks introduced by Watkins [11]. To explain the structure of these graphs, we compose them from smaller parts called networks. A network is an ordered pair (G, U ) that consists of a graph G and a subset U of V (G) whose vertices all have degree 1. The elements of U are the terminal vertices or simply the terminals. The edge incident to a terminal is a terminal edge. Two terminals u and v from the same network or from two disjoint networks, can be joined by adding the edge between the neighbouring vertices of u and v and deleting u and v together with their incident edges. By joining two terminal edges we mean joining two terminals incident to these edges. Suppose that a graph G or a network N was created by joining some terminals of smaller networks Ni . We will call G the composite graph and N the composite network from networks Ni . Each network Ni from which we can get the graph G or network N is a subnetwork of G or of N. Two subnetworks of G are neighbouring if some vertices from these two subnetworks were joined. When necessary, we will call a network N ′ isomorphic to a network N an N-network. If N ′ is a subnetwork of a composite graph or composite network G, then N ′ is a N-subnetwork of G. A nowhere-zero r-flow and the flow number, ΦR (G, U ), on a network (G, U ), are defined similarly as for a normal graph, except that the conservation condition is not required to hold for the terminal vertices. The excess flow at a vertex v equals the sum of the flow values on the edges exiting v minus the sum of the flow values on the edges entering v . Clearly, nonterminal vertices must have excess flow zero. If a terminal edge e is incident only with one terminal u, the excess flow at e is the excess flow at u. Generalized Blanuša snarks are constructed from three different networks: network B, network I and network X (Fig. 2). The terminal edges a, c and b, d are opposite. (1)
The i-th generalized Blanuša snark of type 1, denoted by Bi , is created by combining one copy of network I with i copies of network B. To distinguish among the copies we will use the upper indices. Join the edges as follows: for 1 ≤ k ≤ i − 1, join c k with ak+1 and join dk and bk+1 . Also join the following pairs: (c i , aI ), (c I , a1 ), (di , bI ), (dI , b1 ).
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Fig. 2. Networks used to create Blanuša snarks.
The critical property of network B is that in every 3-edge-colouring of it, the edges a and c have the same colour. The (1) same holds for the edges b and d. This would mean that in Bi the edge c I would need to have the same colour as the edge aI , which is clearly not possible. Therefore the i-th generalized Blanuša snark of type 1 is not 3-edge-colourable. (2) The i-th generalized Blanuša snark of type 2, denoted by Bi , is created similarly. Instead of I we use X and combine it with i − 1 copies of B in the same manner as for the generalized Blanuša snark of type 1. Note that X has a similar property as I; the edges aX and c X must have different colours in every 3-edge-colouring of X . Thus the generalized Blanuša snarks of type 2 also are not 3-edge-colourable. Note that the first graph in each series is isomorphic to the Petersen graph. The second graphs in these series are the two well-known Blanuša snarks [11]. Balanced valuations as used by Jaeger [2] provide a powerful tool for determining the circular flow number of a certain graph. For cubic graphs we can simplify this notion. Let G be a graph and let φ be a nowhere-zero k-flow on G. If φ contains a negative value on an edge e, then we can negate both the orientation of the edge e and the flow value on the edge e. Applying this to all negative edges yields a positive flow φ ′ , where a positive flow is a flow whose edge weights are all positive. The orientation supporting φ ′ is called the positive orientation of G with respect to the flow φ . For S ⊆ V (G), the boundary of S is the set of edges incident with one vertex from S and one vertex from V (G) − S; it is denoted by δG (S ). Let H be a subgraph of G. Given an oriented graph G and a subset S of V (G), the set of edges exiting S, denoted by S + , is the set of edges from δG (S ) whose initial vertex is from S. The set of edges entering S, denoted by S − , is the set of edges from δG (S ) whose terminal vertex is from S. Suppose that G is a cubic graph and φ is a nowhere-zero r-flow on G. Consider the positive orientation of G with respect to φ . Clearly, G now has two types of vertices: black vertices with two entering and one exiting edge and white vertices with one entering and two exiting edges. The numbers of vertices of the two types are equal. Now let S be a subset of V (G), with m black vertices n white vertices, and s boundary edges. The number of edges induced by S is (3m + 3n − s)/2. Thus |S + | = (s + m − n)/2 and |S − | = (s + n − m)/2. Since the largest possible flow value on an edge is r − 1, we have following conditions for the ratio of number of entering edges to number of exiting edges: If one of the values |S + | or |S − | equals 0, the other value also equals zero; otherwise 1 r −1
≤
s+m−n s+n−m
≤ r − 1.
(1)
These bounds must hold for every subset S of vertices of G. A colouring with this property is an r-balanced colouring. A graph G has a nowhere-zero r-flow, if and only if it has an r-balanced colouring [2]. The r-balanced colouring created from the flow φ as described above is the balanced colouring induced by φ . We also define an r-balanced colouring on a network. It is defined like an r-balanced colouring on a graph, but in a network the terminals are not coloured and cannot be contained in the set S. It is well-known that the circular flow number of the Petersen graph is 5 [9]. It is also known that the circular flow number of the two Blanuša snarks is 4.5 (e.g. from the computer search reported in [7]). It is easy to extend the nowhere-zero 4.5flow on two Blanuša snarks to a nowhere-zero 4.5-flow on generalized Blanuša snarks (Fig. 3). We will prove that this is optimal; the circular flow number of all generalized Blanuša snarks equals 4.5. We start with the examination of possible (4 + ε)-balanced colourings of the generalized Blanuša snarks, where ε < 1/2.
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Fig. 3. Nowhere-zero 4.5-flow on generalized Blanuša snarks of type 1 (top) and type 2 (bottom).
Fig. 4. Forbidden configurations.
Fig. 5. Colouring types of network B that avoid the three-path configuration.
The condition (1) allows us to find many forbidden configurations of colours. The two simplest ones are the three-path configuration and the three-star configuration, presented in Fig. 4. A three-path implies ε ≥ 1, a three-star implies r ≥ 1/2— both contradict ε < 1/2. To manage the large amount of colourings we will divide colourings of the network into colouring types. Every colouring within the same colouring type can be obtained from any other colouring of the same type by applying some automorphism of the network to it, or by switching the colours, or both. We start by examining the possible (4 + ε)-balanced colourings of B-subnetworks of generalized Blanuša snarks (j)
Lemma 1. Let ε < 1/2. In every (4 + ε)-balanced colouring of the generalized Blanuša snark Bi , the colouring of each B(j)
subnetwork of Bi belongs to the colouring type C3 , presented in Fig. 5. Proof. First we find all possible colourings of network B that do not contain the three-path configuration. Without loss of generality assume that b1 is black (vertex labels are as in Fig. 2). Two cases may arise. Suppose first that no vertex in {b1 , b3 , b5 , b7 } has a neighbour of the same colour. With b1 black, this forces b1 , b3 , b5 , b7 black and b2 , b4 , b6 , b8 white. This colouring together with the colouring with colours switched will form the colouring type C1 . On the other hand, suppose some vertex in {b1 , b3 , b5 , b7 } has a neighbour of the same colour. We may assume that it is b1 , with b2 also black. To avoid a three-path b3 , b6 , b8 all must be white and b4 and b7 must be black. The last vertex may be coloured arbitrarily. Colouring b5 black yields the type C2 ; white yields the type C3 . All colouring types are displayed on Fig. 5. Note that either 1. No vertex in {b1 , b3 , b5 , b7 } has a neighbour of the same colour. 2. Two opposite vertices in {b1 , b3 , b5 , b7 } have a neighbour of the same colour. These properties are unchanged by applying automorphism to B. Therefore they are also properties of the corresponding colouring type. We show that only the type C3 can be used to colour the generalized Blanuša snarks. First let us concentrate on the type C2 . The choice of S = {b1 , b2 , b4 , b5 , b6 , b7 , b8 } contradicts the inequality (1). Therefore this type is not possible. Excluding type C1 is harder. Suppose that type C1 is used to colour some B-subnetwork Bj . First we show that if one subnetwork Bj is coloured using type C1 , all its neighbouring B-subnetworks must be coloured using type C1 . This is because the B-subnetwork using type C1 cannot be next to two adjacent vertices of the same colour, as this would yield forbidden configurations as shown in Fig. 6. The only other type, C3 , has two adjacent vertices of the same colour incident to its opposite terminal edges, and therefore two adjacent vertices of the same colour are next to both neighbouring B-subnetworks.
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Fig. 6. Type C1 cannot be next to type C3 .
(1)
Fig. 7. A three-star in the subnetwork of Bk containing the subnetwork I and its two neighbouring B-blocks.
Therefore B-subnetworks coloured with types C1 and C3 cannot be neighbouring. If one subnetwork Bj is coloured using type C1 , then all the subnetworks Bi must be coloured using type C1 . First consider the generalized Blanuša snarks of type 1. Since all networks Bi are coloured using type C1 and the total number of black and white vertices must be the same and the colours of i1 and i2 (see Fig. 2) must be different. There are only two colourings of type C1 , and they only differ by switching the colours. If two B-subnetworks neighbouring I have the same colouring, then there is a three-path containing one vertex from each of the neighbouring B-subnetworks and one vertex from the subnetwork I. If the colourings of the neighbouring B-subnetworks are different, then a three-star can be found as indicated in Fig. 7. For the generalized Blanuša snarks of type 2, the situation is simpler. Since the vertices x1 , x3 , x5 , and x7 cannot have a neighbour of the same colour, the vertices x1 , x2 , . . . , x8 must be coloured with alternating colours. Again, x9 and x10 must have different colours. Either x2 , x9 , and x6 form a three-path, or x4 , x10 , and x8 form a three-path. Therefore we cannot use any colouring of type C1 to colour a B-subnetwork in a (4 + ε)-balanced colouring of a generalized Blanuša snark. The only remaining colouring type is C3 . This concludes the proof of the lemma. From now on we only need to consider the colourings of B having type C3 . We switch to modular flows introduced by [8]. A normalized modular r-flow on a graph is an assignment of directions and real flow values from [1; r /2] so that the flow conservation property holds modulo r. An edge with flow value in the interval [1, 1 + r − 4] is a tight edge. An edge with flow value in the interval [1, 1 + 2(r − 4)] is a semi-tight edge. Let φ be a real nowhere-zero r-flow. We change it to positive r-flow and then for each edge with flow value φ(e) from (r /2; r − 1] we negate the orientation of e and change its flow value to r − φ(e). We obtain a normalized modular r-flow φ ′ . We say that φ ′ is the normalized modular r-flow induced by the circular flow φ . We can also construct a real nowhere-zero r-flow from a normalized modular r-flow [8]. The balanced colouring c induced by a flow φ has a connection with the normalized modular flow φ ′ induced by the same flow φ . If there are two adjacent black vertices x and y in c, then all edges on the boundary of {x, y} are tight, and they are all entering x or y in φ ′ . Similarly, if there are two adjacent white vertices x and y in c, then all edges on the boundary of {x, y} are tight, and they are all exiting x or y in φ ′ . As with nowhere-zero flows, normalized modular flows can also be defined for networks by just dropping the flow conversation condition for the terminals. If we create a composite graph (or network) G by joining some vertices of smaller networks Ni , then a normalized modular flow on G can be composed from normalized modular flows on networks Ni , provided that the flow values and the orientations on terminal edges fit together. Also if we have a normalized modular flow on G, then we may decompose it into normalized modular flows on the networks Ni . A semitight terminal edge e is entering N if it is exiting its terminal. Otherwise we say e is exiting N. A set of terminal edges is oriented consistently if they are either all entering N or all exiting N. We sum up the facts from the investigation of balanced valuations on generalized Blanuša snarks in the concept of viability. A normalized modular r-flow on a B-network is viable if it can be obtained by normalizing a nowhere-zero flow φ (that is also a modular nowhere-zero flow) that induces an r-balanced colouring of type C3 . A normalized modular r-flow on a network is viable if it is viable on all its B-subnetworks. Fig. 8 shows the structure and orientation of tight edges on network B in a normalized modular viable flow according to Lemma 1. The following important observation can be made: Lemma 2. In any viable normalized modular r-flow on a B-network, some two opposite terminal edges are both tight and they are not oriented consistently.
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Fig. 8. Tight edges and their orientation.
The next step is to shed light on the global structure of tight and semi-tight edges on the terminal edges of a sequence of neighbouring B-subnetworks. Let Bk be a network arising from k subsequent copies of B joined as in the generalized Blanuša snarks. Since we will use this observation separately for Blanuša snarks of type 1 and Blanuša snarks of type 2, we record the structure of tight edges in Bk in the following technical lemma. Lemma 3. If φ ′ is a viable normalized modular r-flow on the Bk , then for the edges ai , bi , c i and di , where i ∈ {1 . . . k}, at least one of the following holds. 1. All ai , c i are tight and all bi , di are not tight. Moreover the terminal edges a1 and c k in Bk are not oriented consistently. 2. All bi , di are tight and all ai , c i are not tight. Moreover the terminal edges b1 and dk in Bk are not oriented consistently. 3. All ai , bi , c i and di are semi-tight. Moreover the terminal edges a1 and c k in Bk are not oriented consistently, and the terminal edges b1 and dk in Bk are also not oriented consistently. Proof. Suppose first that no subnetwork Bi contains three tight terminal edges. By Lemma 2, one pair of opposite edges in B1 , say a1 and c1 (the other case is the same), is tight and not oriented consistently. The edge d1 is not tight by our assumption that Bi does not contain three tight edges. Therefore (if k > 1) a2 is tight and b2 is not tight. By Lemma 2, the edge c2 is tight and not oriented consistently with a1 . The edge d2 is not tight by the assumption that B2 contains three tight terminal edges. By induction, we conclude that statement 1 holds. On the other hand, suppose that for some i the subnetwork Bi has at least three tight terminal edges. Suppose that (ai , ci ) is the pair of opposite terminal edges guaranteed by Lemma 2. Moreover, suppose that ai is entering Bi while c i is exiting Bi , and that bi is the third tight terminal edge of Bi . All other cases are symmetric to this one, we can obtain the described case either by applying the automorphism of Bk that takes ai to bi or by applying the automorphism of Bk that takes ai to dk−i+1 or by negating the orientation of all edges in the normalized modular flow or by any combination of these methods. Consider first that bi is entering the subnetwork Bi . The sum of excess flows in the edges ai and bi in Bi is in the interval [0, 2 + 2ε]. From the flow conservation property we have that for every j the sum of excess flows at aj and bj in Bj is in the interval [2, 2 + 2ε], and for every j the sum of excess flows of c j and dj in Bj is in the interval [−2 − 2ε, −2]. In particular, this means that di is semi-tight and is exiting Bi , not oriented consistently to bi . Consider now the subnetwork Bi+1 (if it exists). The edges ai+1 (was joined to c i ) and bi+1 (was joined to di ) are both entering Bi+1 . One is tight and one is semi-tight. Lemma 2 says that either c i+1 or di+1 is tight and is exiting Bi+1 ; say it is the edge c i+1 . The sum of excess flows of c i+1 and di+1 in Bi+1 is in the interval [−2 − 2ε, −2]. Therefore the other edge, di+1 , is semi-tight and is exiting Bi+1 . We can continue in this way for j > i. The same approach can be used for all j < i. Thus statement 3 holds in this case. The argumentation is very similar if bi is exiting the subnetwork Bi . From the flow conservation condition we get that the for every j sum of excess flows of aj and bj in Bj is in the interval [−ε, ε] and the sum of excess flows of c j and dj in Bi is also in the interval [−ε, ε]. Reasoning as in previous case, this implies that the statement 3 holds. Now we can show that the generalized Blanuša snarks of type 1 cannot have a nowhere-zero (4 + ε)-flow, for ε < 1/2. Lemma 4. The circular flow number of the generalized Blanuša snarks of type 1 is at least 4 + 1/2. (1)
Proof. Let Bi be a Blanuša snark of type 1. Suppose, for contradiction, that it has a circular nowhere-zero (4 + ε)-flow φ for ε < 1/2. Let φ ′ be a normalized modular (4 + ε)-flow induced by φ . According to the Lemma 1, φ ′ is viable. By Lemma 3 at least one of the pairs (aI , c I ) and (bI , dI ) contains two semitight edges not oriented consistently. But then the flow value on the edge i1 i2 is in [−2ε, 2ε]. This contradicts that φ ′ is nowhere-zero. The final argument for the generalized Blanuša snarks of type 2 will be slightly more complicated. Lemma 5. The circular flow number of the generalized Blanuša snarks of type 2 is at least 4 + 1/2. (2)
Proof. Let Bi
be a Blanuša snark of type 2. If it has a circular nowhere-zero (4 + ε)-flow φ for ε < 1/2, then there is also (2)
a (4 + ε)-balanced colouring of Bi . Let us consider the possible (4 + ε)-balanced colourings of network X , for ε < 1/2.
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Fig. 9. Colouring types of network X .
Fig. 10. Tight edge structure of network X .
(2)
There is the same number of black and white vertices in all B-subnetworks of Bi , therefore the number of black and white vertices must be the same also in X . The vertices x1 , x2 , . . . , x8 cannot be coloured alternately—the vertices x9 and x10 need to have different colours and either x2 , x9 , and x6 form a three-path, or x4 , x10 , and x8 form a three-path. Therefore without loss of generality we may suppose that both x1 and x2 are black. A simple case analysis shows that there are three possible colouring types C4 , C5 , and C6 , presented in Fig. 9. If φ induces colouring of X of type C5 , then two opposite edges in X are tight and oriented consistently. This contradicts Lemma 3. If φ induces colouring of X of type C4 or C6 only statement 3 of Lemma 3 can hold. Therefore the dashed edges from Fig. 10 are semitight and oriented as in the figure. In both cases the flow conservation condition cannot hold on the vertex incident to a dashed edge. Therefore the circular flow number of the generalized Blanuša snarks of type 2 is at least 4.5. Lemmas 4 and 5 show, that the circular flow number of the generalized Blanuša snarks is at least 4.5. Nowhere-zero 4.5flows on the generalized Blanuša snarks other than the Petersen graph are presented in Fig. 3. This shows that the circular flow number of generalized Blanuša snarks is 4.5. Acknowledgments I would like to thank Edita Máčajová for determining the circular flow number of small generalized Blanuša snarks by computer, and Mirko Horňák, Martin Škoviera, the referees, and the editor for useful comments which helped to improve the text. The author acknowledges partial support from the research grants APVV-0223-10, VEGA 1/0634/09, and from the APVV grant ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation. References [1] L.A. Goddyn, M. Tarsi, C. Zhang, On (k, d)-colorings and fractional nowhere-zero flows, J. Graph Theory 28 (1998) 155–161. [2] F. Jaeger, Balanced valuations and flows in multigraphs, Proc. Amer. Math. Soc. 55 (1975) 237–242. [3] F. Jaeger, Nowhere-zero flow problems, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, Vol. 3, Academic Press, London, 1988, pp. 71–95. [4] M. Kochol, Smallest counterexample to the 5-flow conjecture has girth at least eleven, J. Combin. Theory Ser. B 100 (2010) 381–389. [5] R. Lukot’ka, M. Škoviera, Real flow number and the cycle rank of a graph, J. Graph Theory 59 (2009) 11–16. [6] R. Lukot’ka, M. Škoviera, Snarks with given real flow number, J. Graph Theory 68 (2011) 189–201. [7] E. Máčajová, A. Raspaud, On the strong circular 5-flow conjecture, J. Graph Theory 52 (2006) 307–316. [8] Z. Pan, X. Zhu, Construction of graphs with given circular flow numbers, J. Graph Theory 43 (2003) 304–318. [9] E. Steffen, Circular flow numbers of regular multigraphs, J. Graph Theory 36 (2001) 24–34. [10] W.T. Tutte, On the imbedding of linear graphs in surfaces, Proc. Lond. Math. Soc. (2) 51 (1949) 464–483. [11] J.J. Watkins, Snarks, Ann. New York Acad. Sci. 576 (1989) 606–622.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Circular flow number of generalized Blanuša snarks Robert Lukot’ka Department of Mathematics and Computer Science, Trnava University in Trnava, Trnava, 918 43, Slovakia
article
info
Article history: Received 17 May 2011 Received in revised form 9 January 2013 Accepted 14 January 2013 Available online 8 February 2013 Keywords: Circular flow number Real flow number Generalized Blanuša snarks
abstract A circular flow on a graph is an assignment of directions and flow values from R to the edges so that for each vertex the sum of the flow values on exiting edges equals the sum of the flow values on entering edges. A circular nowhere-zero r-flow is a circular flow φ with flow values satisfying 1 ≤ |φ(e)| ≤ r − 1 for each edge e. The circular flow number of a graph G is the infimum of all reals r such that G has a circular nowhere-zero r-flow. We prove that the circular flow number of all generalized Blanuša snarks except for the Petersen graph is 4.5. We also bound the circular flow number of Goldberg snarks, both from above and from below. © 2013 Elsevier B.V. All rights reserved.
1. Introduction and preliminaries A k-flow is an assignment of directions and integer flow values to the edges so that the flow conservation condition is satisfied. The flow conservation condition asserts that for each vertex the sum of the flow values on exiting edges equals the sum of the flow values on entering edges. A nowhere-zero k-flow is a k-flow that does not use value 0. The concept of nowhere-zero flows can be extended to the real numbers. This extension can be done in multiple different ways; nevertheless, all the approaches suggested by various authors [1,3,9] turned out to be equivalent. Perhaps the most natural way to introduce the real numbers to nowhere-zero flows is to directly employ real numbers instead of integers as the flow values. Let r ≥ 2 be a real number. A circular nowhere-zero r-flow φ on a graph G is an assignment of directions and real flow values from [1; r − 1] to the edges of G so that the flow conservation condition holds. The circular flow number of a graph G (also known as the real flow number [5,6]) is the infimum of all reals r such that G has a circular nowhere-zero r-flow. This infimum is attained and it is rational [1]. Moreover, the ordinary flow number is the ceiling of the circular flow number [1]. The exact value of the circular flow number is only known for few classes of graphs. This includes e.g. eulerian, regular bipartite, and complete graphs [9]. If we concentrate on cubic graphs, the situation however is much more clearer, at least for 3-edge-colourable graphs. Every bipartite 3-edge-colourable graph has nowhere-zero 3-flow. According to Steffen’s gap theorem [9], every cubic graph with circular flow number strictly less than 4 is bipartite. Since 3-edge-colourable graphs have a nowhere-zero 4-flow, a non-bipartite 3-edge-colourable graph has circular flow number 4. However, only very little is known about the circular flow number of non-3-edge-colourable cubic graphs. A bridgeless non-3-edge-colourable cubic graph is called a snark. Snarks are an intensively studied class of graphs, probably due to the fact that for many important conjectures in graph theory, any minimal counterexample must be a snark. One of these conjectures is Tutte’s 5-flow Conjecture—every bridgeless graph has a nowhere-zero 5-flow [10]. Indeed, Kochol [4] showed that any minimal counterexample to Tutte’s 5-flow Conjecture must be a cyclically 6-edge-connected snark of girth at least 11. Recent results answer several questions about the circular flow number of snarks [5–7]. However, these result use very specific graphs and tell us very little about how to tackle the problem of determining the circular flow number of a given
E-mail address: [email protected]. 0012-365X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2013.01.010
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Fig. 1. A nowhere-zero (4 + 1/k)-flow on the Goldberg snark Hk .
snark. To develop new methods we chose well-known infinite families of snarks with a simple global structure. Perhaps the best example of such snarks are the Isaacs snarks, whose circular flow number was determined in [5]. The main objective of this article is to determine the circular flow number of generalized Blanuša snarks, which are two families of snarks with a similar ‘‘circular’’ structure. As the main result of this paper, we show that the circular flow number of all generalized Blanuša snarks except for the Petersen graph is 4.5. Moreover, we bound the circular flow number of graphs from another well-known class of snarks, the Goldberg snarks. Goldberg snarks are constructed like the well-known Isaacs snarks. Instead of a Y -shaped block we use the block depicted in Fig. 1 bottom. We found a nowhere-zero (4 + 1/k)-flow on the Goldberg snark G2k+1 , similar to a nowhere-zero (4 + 1/k)flow on the Isaacs snark I2k+1 (Fig. 1). Together with a natural lower bound for the circular flow number of a snark [5] we know that the circular flow number of the Goldberg snark G2k+1 is in the interval [4 + 1/(2k + 1); 4 + 1/k].
2. Generalized Blanuša snarks The generalized Blanuša snarks of type 1 and type 2 are two important classes of snarks introduced by Watkins [11]. To explain the structure of these graphs, we compose them from smaller parts called networks. A network is an ordered pair (G, U ) that consists of a graph G and a subset U of V (G) whose vertices all have degree 1. The elements of U are the terminal vertices or simply the terminals. The edge incident to a terminal is a terminal edge. Two terminals u and v from the same network or from two disjoint networks, can be joined by adding the edge between the neighbouring vertices of u and v and deleting u and v together with their incident edges. By joining two terminal edges we mean joining two terminals incident to these edges. Suppose that a graph G or a network N was created by joining some terminals of smaller networks Ni . We will call G the composite graph and N the composite network from networks Ni . Each network Ni from which we can get the graph G or network N is a subnetwork of G or of N. Two subnetworks of G are neighbouring if some vertices from these two subnetworks were joined. When necessary, we will call a network N ′ isomorphic to a network N an N-network. If N ′ is a subnetwork of a composite graph or composite network G, then N ′ is a N-subnetwork of G. A nowhere-zero r-flow and the flow number, ΦR (G, U ), on a network (G, U ), are defined similarly as for a normal graph, except that the conservation condition is not required to hold for the terminal vertices. The excess flow at a vertex v equals the sum of the flow values on the edges exiting v minus the sum of the flow values on the edges entering v . Clearly, nonterminal vertices must have excess flow zero. If a terminal edge e is incident only with one terminal u, the excess flow at e is the excess flow at u. Generalized Blanuša snarks are constructed from three different networks: network B, network I and network X (Fig. 2). The terminal edges a, c and b, d are opposite. (1)
The i-th generalized Blanuša snark of type 1, denoted by Bi , is created by combining one copy of network I with i copies of network B. To distinguish among the copies we will use the upper indices. Join the edges as follows: for 1 ≤ k ≤ i − 1, join c k with ak+1 and join dk and bk+1 . Also join the following pairs: (c i , aI ), (c I , a1 ), (di , bI ), (dI , b1 ).
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Fig. 2. Networks used to create Blanuša snarks.
The critical property of network B is that in every 3-edge-colouring of it, the edges a and c have the same colour. The (1) same holds for the edges b and d. This would mean that in Bi the edge c I would need to have the same colour as the edge aI , which is clearly not possible. Therefore the i-th generalized Blanuša snark of type 1 is not 3-edge-colourable. (2) The i-th generalized Blanuša snark of type 2, denoted by Bi , is created similarly. Instead of I we use X and combine it with i − 1 copies of B in the same manner as for the generalized Blanuša snark of type 1. Note that X has a similar property as I; the edges aX and c X must have different colours in every 3-edge-colouring of X . Thus the generalized Blanuša snarks of type 2 also are not 3-edge-colourable. Note that the first graph in each series is isomorphic to the Petersen graph. The second graphs in these series are the two well-known Blanuša snarks [11]. Balanced valuations as used by Jaeger [2] provide a powerful tool for determining the circular flow number of a certain graph. For cubic graphs we can simplify this notion. Let G be a graph and let φ be a nowhere-zero k-flow on G. If φ contains a negative value on an edge e, then we can negate both the orientation of the edge e and the flow value on the edge e. Applying this to all negative edges yields a positive flow φ ′ , where a positive flow is a flow whose edge weights are all positive. The orientation supporting φ ′ is called the positive orientation of G with respect to the flow φ . For S ⊆ V (G), the boundary of S is the set of edges incident with one vertex from S and one vertex from V (G) − S; it is denoted by δG (S ). Let H be a subgraph of G. Given an oriented graph G and a subset S of V (G), the set of edges exiting S, denoted by S + , is the set of edges from δG (S ) whose initial vertex is from S. The set of edges entering S, denoted by S − , is the set of edges from δG (S ) whose terminal vertex is from S. Suppose that G is a cubic graph and φ is a nowhere-zero r-flow on G. Consider the positive orientation of G with respect to φ . Clearly, G now has two types of vertices: black vertices with two entering and one exiting edge and white vertices with one entering and two exiting edges. The numbers of vertices of the two types are equal. Now let S be a subset of V (G), with m black vertices n white vertices, and s boundary edges. The number of edges induced by S is (3m + 3n − s)/2. Thus |S + | = (s + m − n)/2 and |S − | = (s + n − m)/2. Since the largest possible flow value on an edge is r − 1, we have following conditions for the ratio of number of entering edges to number of exiting edges: If one of the values |S + | or |S − | equals 0, the other value also equals zero; otherwise 1 r −1
≤
s+m−n s+n−m
≤ r − 1.
(1)
These bounds must hold for every subset S of vertices of G. A colouring with this property is an r-balanced colouring. A graph G has a nowhere-zero r-flow, if and only if it has an r-balanced colouring [2]. The r-balanced colouring created from the flow φ as described above is the balanced colouring induced by φ . We also define an r-balanced colouring on a network. It is defined like an r-balanced colouring on a graph, but in a network the terminals are not coloured and cannot be contained in the set S. It is well-known that the circular flow number of the Petersen graph is 5 [9]. It is also known that the circular flow number of the two Blanuša snarks is 4.5 (e.g. from the computer search reported in [7]). It is easy to extend the nowhere-zero 4.5flow on two Blanuša snarks to a nowhere-zero 4.5-flow on generalized Blanuša snarks (Fig. 3). We will prove that this is optimal; the circular flow number of all generalized Blanuša snarks equals 4.5. We start with the examination of possible (4 + ε)-balanced colourings of the generalized Blanuša snarks, where ε < 1/2.
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Fig. 3. Nowhere-zero 4.5-flow on generalized Blanuša snarks of type 1 (top) and type 2 (bottom).
Fig. 4. Forbidden configurations.
Fig. 5. Colouring types of network B that avoid the three-path configuration.
The condition (1) allows us to find many forbidden configurations of colours. The two simplest ones are the three-path configuration and the three-star configuration, presented in Fig. 4. A three-path implies ε ≥ 1, a three-star implies r ≥ 1/2— both contradict ε < 1/2. To manage the large amount of colourings we will divide colourings of the network into colouring types. Every colouring within the same colouring type can be obtained from any other colouring of the same type by applying some automorphism of the network to it, or by switching the colours, or both. We start by examining the possible (4 + ε)-balanced colourings of B-subnetworks of generalized Blanuša snarks (j)
Lemma 1. Let ε < 1/2. In every (4 + ε)-balanced colouring of the generalized Blanuša snark Bi , the colouring of each B(j)
subnetwork of Bi belongs to the colouring type C3 , presented in Fig. 5. Proof. First we find all possible colourings of network B that do not contain the three-path configuration. Without loss of generality assume that b1 is black (vertex labels are as in Fig. 2). Two cases may arise. Suppose first that no vertex in {b1 , b3 , b5 , b7 } has a neighbour of the same colour. With b1 black, this forces b1 , b3 , b5 , b7 black and b2 , b4 , b6 , b8 white. This colouring together with the colouring with colours switched will form the colouring type C1 . On the other hand, suppose some vertex in {b1 , b3 , b5 , b7 } has a neighbour of the same colour. We may assume that it is b1 , with b2 also black. To avoid a three-path b3 , b6 , b8 all must be white and b4 and b7 must be black. The last vertex may be coloured arbitrarily. Colouring b5 black yields the type C2 ; white yields the type C3 . All colouring types are displayed on Fig. 5. Note that either 1. No vertex in {b1 , b3 , b5 , b7 } has a neighbour of the same colour. 2. Two opposite vertices in {b1 , b3 , b5 , b7 } have a neighbour of the same colour. These properties are unchanged by applying automorphism to B. Therefore they are also properties of the corresponding colouring type. We show that only the type C3 can be used to colour the generalized Blanuša snarks. First let us concentrate on the type C2 . The choice of S = {b1 , b2 , b4 , b5 , b6 , b7 , b8 } contradicts the inequality (1). Therefore this type is not possible. Excluding type C1 is harder. Suppose that type C1 is used to colour some B-subnetwork Bj . First we show that if one subnetwork Bj is coloured using type C1 , all its neighbouring B-subnetworks must be coloured using type C1 . This is because the B-subnetwork using type C1 cannot be next to two adjacent vertices of the same colour, as this would yield forbidden configurations as shown in Fig. 6. The only other type, C3 , has two adjacent vertices of the same colour incident to its opposite terminal edges, and therefore two adjacent vertices of the same colour are next to both neighbouring B-subnetworks.
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Fig. 6. Type C1 cannot be next to type C3 .
(1)
Fig. 7. A three-star in the subnetwork of Bk containing the subnetwork I and its two neighbouring B-blocks.
Therefore B-subnetworks coloured with types C1 and C3 cannot be neighbouring. If one subnetwork Bj is coloured using type C1 , then all the subnetworks Bi must be coloured using type C1 . First consider the generalized Blanuša snarks of type 1. Since all networks Bi are coloured using type C1 and the total number of black and white vertices must be the same and the colours of i1 and i2 (see Fig. 2) must be different. There are only two colourings of type C1 , and they only differ by switching the colours. If two B-subnetworks neighbouring I have the same colouring, then there is a three-path containing one vertex from each of the neighbouring B-subnetworks and one vertex from the subnetwork I. If the colourings of the neighbouring B-subnetworks are different, then a three-star can be found as indicated in Fig. 7. For the generalized Blanuša snarks of type 2, the situation is simpler. Since the vertices x1 , x3 , x5 , and x7 cannot have a neighbour of the same colour, the vertices x1 , x2 , . . . , x8 must be coloured with alternating colours. Again, x9 and x10 must have different colours. Either x2 , x9 , and x6 form a three-path, or x4 , x10 , and x8 form a three-path. Therefore we cannot use any colouring of type C1 to colour a B-subnetwork in a (4 + ε)-balanced colouring of a generalized Blanuša snark. The only remaining colouring type is C3 . This concludes the proof of the lemma. From now on we only need to consider the colourings of B having type C3 . We switch to modular flows introduced by [8]. A normalized modular r-flow on a graph is an assignment of directions and real flow values from [1; r /2] so that the flow conservation property holds modulo r. An edge with flow value in the interval [1, 1 + r − 4] is a tight edge. An edge with flow value in the interval [1, 1 + 2(r − 4)] is a semi-tight edge. Let φ be a real nowhere-zero r-flow. We change it to positive r-flow and then for each edge with flow value φ(e) from (r /2; r − 1] we negate the orientation of e and change its flow value to r − φ(e). We obtain a normalized modular r-flow φ ′ . We say that φ ′ is the normalized modular r-flow induced by the circular flow φ . We can also construct a real nowhere-zero r-flow from a normalized modular r-flow [8]. The balanced colouring c induced by a flow φ has a connection with the normalized modular flow φ ′ induced by the same flow φ . If there are two adjacent black vertices x and y in c, then all edges on the boundary of {x, y} are tight, and they are all entering x or y in φ ′ . Similarly, if there are two adjacent white vertices x and y in c, then all edges on the boundary of {x, y} are tight, and they are all exiting x or y in φ ′ . As with nowhere-zero flows, normalized modular flows can also be defined for networks by just dropping the flow conversation condition for the terminals. If we create a composite graph (or network) G by joining some vertices of smaller networks Ni , then a normalized modular flow on G can be composed from normalized modular flows on networks Ni , provided that the flow values and the orientations on terminal edges fit together. Also if we have a normalized modular flow on G, then we may decompose it into normalized modular flows on the networks Ni . A semitight terminal edge e is entering N if it is exiting its terminal. Otherwise we say e is exiting N. A set of terminal edges is oriented consistently if they are either all entering N or all exiting N. We sum up the facts from the investigation of balanced valuations on generalized Blanuša snarks in the concept of viability. A normalized modular r-flow on a B-network is viable if it can be obtained by normalizing a nowhere-zero flow φ (that is also a modular nowhere-zero flow) that induces an r-balanced colouring of type C3 . A normalized modular r-flow on a network is viable if it is viable on all its B-subnetworks. Fig. 8 shows the structure and orientation of tight edges on network B in a normalized modular viable flow according to Lemma 1. The following important observation can be made: Lemma 2. In any viable normalized modular r-flow on a B-network, some two opposite terminal edges are both tight and they are not oriented consistently.
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Fig. 8. Tight edges and their orientation.
The next step is to shed light on the global structure of tight and semi-tight edges on the terminal edges of a sequence of neighbouring B-subnetworks. Let Bk be a network arising from k subsequent copies of B joined as in the generalized Blanuša snarks. Since we will use this observation separately for Blanuša snarks of type 1 and Blanuša snarks of type 2, we record the structure of tight edges in Bk in the following technical lemma. Lemma 3. If φ ′ is a viable normalized modular r-flow on the Bk , then for the edges ai , bi , c i and di , where i ∈ {1 . . . k}, at least one of the following holds. 1. All ai , c i are tight and all bi , di are not tight. Moreover the terminal edges a1 and c k in Bk are not oriented consistently. 2. All bi , di are tight and all ai , c i are not tight. Moreover the terminal edges b1 and dk in Bk are not oriented consistently. 3. All ai , bi , c i and di are semi-tight. Moreover the terminal edges a1 and c k in Bk are not oriented consistently, and the terminal edges b1 and dk in Bk are also not oriented consistently. Proof. Suppose first that no subnetwork Bi contains three tight terminal edges. By Lemma 2, one pair of opposite edges in B1 , say a1 and c1 (the other case is the same), is tight and not oriented consistently. The edge d1 is not tight by our assumption that Bi does not contain three tight edges. Therefore (if k > 1) a2 is tight and b2 is not tight. By Lemma 2, the edge c2 is tight and not oriented consistently with a1 . The edge d2 is not tight by the assumption that B2 contains three tight terminal edges. By induction, we conclude that statement 1 holds. On the other hand, suppose that for some i the subnetwork Bi has at least three tight terminal edges. Suppose that (ai , ci ) is the pair of opposite terminal edges guaranteed by Lemma 2. Moreover, suppose that ai is entering Bi while c i is exiting Bi , and that bi is the third tight terminal edge of Bi . All other cases are symmetric to this one, we can obtain the described case either by applying the automorphism of Bk that takes ai to bi or by applying the automorphism of Bk that takes ai to dk−i+1 or by negating the orientation of all edges in the normalized modular flow or by any combination of these methods. Consider first that bi is entering the subnetwork Bi . The sum of excess flows in the edges ai and bi in Bi is in the interval [0, 2 + 2ε]. From the flow conservation property we have that for every j the sum of excess flows at aj and bj in Bj is in the interval [2, 2 + 2ε], and for every j the sum of excess flows of c j and dj in Bj is in the interval [−2 − 2ε, −2]. In particular, this means that di is semi-tight and is exiting Bi , not oriented consistently to bi . Consider now the subnetwork Bi+1 (if it exists). The edges ai+1 (was joined to c i ) and bi+1 (was joined to di ) are both entering Bi+1 . One is tight and one is semi-tight. Lemma 2 says that either c i+1 or di+1 is tight and is exiting Bi+1 ; say it is the edge c i+1 . The sum of excess flows of c i+1 and di+1 in Bi+1 is in the interval [−2 − 2ε, −2]. Therefore the other edge, di+1 , is semi-tight and is exiting Bi+1 . We can continue in this way for j > i. The same approach can be used for all j < i. Thus statement 3 holds in this case. The argumentation is very similar if bi is exiting the subnetwork Bi . From the flow conservation condition we get that the for every j sum of excess flows of aj and bj in Bj is in the interval [−ε, ε] and the sum of excess flows of c j and dj in Bi is also in the interval [−ε, ε]. Reasoning as in previous case, this implies that the statement 3 holds. Now we can show that the generalized Blanuša snarks of type 1 cannot have a nowhere-zero (4 + ε)-flow, for ε < 1/2. Lemma 4. The circular flow number of the generalized Blanuša snarks of type 1 is at least 4 + 1/2. (1)
Proof. Let Bi be a Blanuša snark of type 1. Suppose, for contradiction, that it has a circular nowhere-zero (4 + ε)-flow φ for ε < 1/2. Let φ ′ be a normalized modular (4 + ε)-flow induced by φ . According to the Lemma 1, φ ′ is viable. By Lemma 3 at least one of the pairs (aI , c I ) and (bI , dI ) contains two semitight edges not oriented consistently. But then the flow value on the edge i1 i2 is in [−2ε, 2ε]. This contradicts that φ ′ is nowhere-zero. The final argument for the generalized Blanuša snarks of type 2 will be slightly more complicated. Lemma 5. The circular flow number of the generalized Blanuša snarks of type 2 is at least 4 + 1/2. (2)
Proof. Let Bi
be a Blanuša snark of type 2. If it has a circular nowhere-zero (4 + ε)-flow φ for ε < 1/2, then there is also (2)
a (4 + ε)-balanced colouring of Bi . Let us consider the possible (4 + ε)-balanced colourings of network X , for ε < 1/2.
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Fig. 9. Colouring types of network X .
Fig. 10. Tight edge structure of network X .
(2)
There is the same number of black and white vertices in all B-subnetworks of Bi , therefore the number of black and white vertices must be the same also in X . The vertices x1 , x2 , . . . , x8 cannot be coloured alternately—the vertices x9 and x10 need to have different colours and either x2 , x9 , and x6 form a three-path, or x4 , x10 , and x8 form a three-path. Therefore without loss of generality we may suppose that both x1 and x2 are black. A simple case analysis shows that there are three possible colouring types C4 , C5 , and C6 , presented in Fig. 9. If φ induces colouring of X of type C5 , then two opposite edges in X are tight and oriented consistently. This contradicts Lemma 3. If φ induces colouring of X of type C4 or C6 only statement 3 of Lemma 3 can hold. Therefore the dashed edges from Fig. 10 are semitight and oriented as in the figure. In both cases the flow conservation condition cannot hold on the vertex incident to a dashed edge. Therefore the circular flow number of the generalized Blanuša snarks of type 2 is at least 4.5. Lemmas 4 and 5 show, that the circular flow number of the generalized Blanuša snarks is at least 4.5. Nowhere-zero 4.5flows on the generalized Blanuša snarks other than the Petersen graph are presented in Fig. 3. This shows that the circular flow number of generalized Blanuša snarks is 4.5. Acknowledgments I would like to thank Edita Máčajová for determining the circular flow number of small generalized Blanuša snarks by computer, and Mirko Horňák, Martin Škoviera, the referees, and the editor for useful comments which helped to improve the text. The author acknowledges partial support from the research grants APVV-0223-10, VEGA 1/0634/09, and from the APVV grant ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation. References [1] L.A. Goddyn, M. Tarsi, C. Zhang, On (k, d)-colorings and fractional nowhere-zero flows, J. Graph Theory 28 (1998) 155–161. [2] F. Jaeger, Balanced valuations and flows in multigraphs, Proc. Amer. Math. Soc. 55 (1975) 237–242. [3] F. Jaeger, Nowhere-zero flow problems, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, Vol. 3, Academic Press, London, 1988, pp. 71–95. [4] M. Kochol, Smallest counterexample to the 5-flow conjecture has girth at least eleven, J. Combin. Theory Ser. B 100 (2010) 381–389. [5] R. Lukot’ka, M. Škoviera, Real flow number and the cycle rank of a graph, J. Graph Theory 59 (2009) 11–16. [6] R. Lukot’ka, M. Škoviera, Snarks with given real flow number, J. Graph Theory 68 (2011) 189–201. [7] E. Máčajová, A. Raspaud, On the strong circular 5-flow conjecture, J. Graph Theory 52 (2006) 307–316. [8] Z. Pan, X. Zhu, Construction of graphs with given circular flow numbers, J. Graph Theory 43 (2003) 304–318. [9] E. Steffen, Circular flow numbers of regular multigraphs, J. Graph Theory 36 (2001) 24–34. [10] W.T. Tutte, On the imbedding of linear graphs in surfaces, Proc. Lond. Math. Soc. (2) 51 (1949) 464–483. [11] J.J. Watkins, Snarks, Ann. New York Acad. Sci. 576 (1989) 606–622.