Diskrete Mathematik und Optimierung

Transcription

1 Diskrete Mathematik und Optimierung Winfried Hochstättler: A flow theory for the dichromatic number (extended version of feu-dmo ) Technical Report feu-dmo Contact: winfried.hochstaettler@fernuni-hagen.de FernUniversität in Hagen Fakultät für Mathematik und Informatik Lehrgebiet für Diskrete Mathematik und Optimierung D Hagen

2 2000 Mathematics Subject Classification: 05C17, 05C20, 05C15 Keywords: dichromatic number, colorings, flows, oriented

3 A flow theory for the dichromatic number Winfried Hochstättler FernUniversität in Hagen, Fakultät für Mathematik und Informatik Universitätsstr. 1, Hagen, Germany Abstract We transfer Tutte s theory for analyzing the chromatic number of a graph using nowhere-zero-coflows and -flows (NZ-flows) to the dichromatic number of a digraph and define Neumann-Lara-flows (NL-flows). We prove that any digraph whose underlying (multi-)graph is 3-edgeconnected admits a NL-3-flow, and even a NL-2-flow in case the underlying graph is 4-edge connected. We conjecture that 3-edge-connectivity already guarantees the existence of a NL-2-flow, which, if true, would imply the 2-Color-Conjecture for planar graphs due to Víctor Neumann- Lara. Finally we present an extension of the theory to oriented matroids. 1 Introduction Víctor Neumann-Lara [12] introduced the dichromatic number χ(d) of a digraph D as the smallest integer k such that the vertices V of D can be colored with k colors and each color class induces a directed acyclic graph. This is a proper generalization of the chromatic number: if D is a digraph, with underlying graph G, where for every arc there exists an antiparallel one, then clearly χ(g) = χ(d). The purpose of this paper is to give a characterization of the dichromatic number in terms of coflows of the digraph and to develop a flow theory dual to this. Neumann-Lara conjectured that the dichromatic number of an orientation of a simple planar graph is bounded by 2. Conjecture 1 (Neumann-Lara [13]). If D is the orientation of a simple planar graph, then χ(d) 2. Using planar duality this 2-Color-Conjecture (2CC) is equivalent to a bound of 2 on a certain flow, which we will call NL-flow. The non-existence of antiparallel arcs in the planar graph dualizes to 3-edge-connectivity of the underlying (multi-)graph. We will show that, in contrast to the classical case of 4-coloring of planar graphs where the Petersen graph is an obstruction to the existence of a NZ-4-flow [17], any orientation of the Petersen graph admits a NL-2-flow. If one could show that a NL-2-flow exists in any 3-edge-connected-digraph this would verify the 2-Color-Conjecture. 1

4 Using techniques from [10] we can show that 4-edge connectivity of the underlying graph is sufficient for the existence of a NL-2-flow, while we can only prove the existence of a NL-4-flow in the 3-edge-connected case. Applying a result of Seymour [15] Matt DeVos [3] helped us to improve this and verify the existence of a NL-3-flow in the 3-edge-connected case. Our notation is fairly standard and, if not explicitely defined, should follow [4] for graphs and [1] for oriented matroids. Note that all our digraphs may have parallel and antiparallel arcs. A weak cycle in a digraph D is a not necessarily directed cycle. 2 Neumann-Lara-flows and -coflows We first recall Tutte s definition of a NZ-k-flow. Let G = (V, E) be a bridgeless graph. A nowhere-zero-k-flow (D, φ), or a NZ-k-flow for short is an orientation D of G together with a map φ : {±1,..., ±(k 1)} satisfying Kirchhoff s law of flow conservation f(a) = f(a) for all v V. a δ (v) a δ + (v) We generalize this to digraphs as follows: Let D = (V, A) be a directed graph. A mapping f = (f 1, f 2 ) : A Z 2 is called a Neumann-Lara-flow or NL-flow for short, if both components of f satisfy Kirchhoff s law of flow conservation and, whenever f 1 (a) = 0, for some arc a A, then necessarily f 2 (a) > 0. A NL-flow is a NL-k-flow, if additionally f 1 (a) < k for all a A. Note that the existence of a NL-k-flow may depend on the orientation of the digraph D, e.g. a connected digraph D has a NL-1-flow if and only if it is strongly connected. A mapping f = (f1, f2 ) : A Z 2 is a Neumann-Lara-coflow, a NLcoflow for short, if it satisfies Kirchhoff s law for cycles, i.e. for each weak cycle C of D fi (a) = fi (a), (1) a C + a C where C + and C denote the arcs of C that are traversed in forward resp. backward direction and, whenever f 1 (a) = 0, for some arc a A, then necessarily f 2 (a) > 0. A NL-coflow is a NL-k-coflow, if in addition f 1 (a) < k for all a A. The following theorem was our motivation to define and study NL-flows. 2

5 Theorem 1. Let D = (V, A) be a loopless directed graph. Then D has a NL-kcoflow if and only if it has dichromatic number at most k. Proof. Clearly, we may assume that D is connected. Let f : A Z 2 be a NLk-coflow. We define a coloring of c using the colors {0, 1,..., k 1} as follows. Choose an arbitrary vertex v V which receives color zero c(v) = 0. Now let w be another vertex and P 1 be a (not necessarily directed) v-w-path in D. Then we define the preliminary color c(w) of w as c(w) = f1 (a) f1 (a), a P + 1 a P 1 where P 1 + and P1 denote the arcs of P 1 that are traversed in positive resp. negative direction and claim that this value is independent of the chosen path. Namely, if P 2 is another such path, then the concatenation of P 1 traversed forwards and P 2 traversed backwards is a closed tour and hence can be decomposed into circuits of D. Since f1 is a coflow in D, f1 sums to zero on all of these circuits. Hence f1 (a) f1 (a) f1 (a) + f1 (a) = 0 a P + 1 a P 1 a P + 2 a P 2 and c is well defined. To get c in the proper range we reduce it modulo k, i.e. we set c(w) = c(w) mod k. We are left to verify that the color classes of this coloring induce acyclic subdigraphs. Assume we had a directed cycle in one color class. Then f1 (a) = 0 on each arc a C implying f2 (a) > 0, hence f2 (a) > 0 = f2 (a) = f2 (a) a C + a C a contradicting the definition of a coflow. On the other hand, if we have a coloring c with colors {0,..., k 1} such that each color class induces an acyclic directed graph, we define a NL-k-coflow as follows. If a = (v, w) A is an arc of D we put f 1 (a) = c(w) c(v). Since f 1 this way is defined by a potential it vanishes on every cycle and hence satisfies (1). Let A 1 denote the set of arcs which receive non-zero f 1 value A 1 := {a A f 1 (a) 0}. Since each color class induces an acyclic directed graph, already D \ A 1 must be acyclic. Hence, using topological sorting we find an ordering v 1,..., v n of the vertices of V such that for each a = (v j, v k ) A \ A 1 we have j < k. Hence putting n 1 f2 = 1 ({v1,...,v i}), i=1 3

6 where 1 if j i < k, 1 ({v1,...,v i})((v j, v k )) := 1 if k i < j, 0 otherwise denotes the directed characteristic function of the cut defined by {v 1,..., v i }, we find a function that vanishes on all cycles and is strictly positive on A \ A 1. Note that, for a = (v i, v j ) A, we get f 2 (a) = j i. Since both, f 1 and f 2, have a potential, they satisfy (1). 3 Planar Digraphs We call a (multi)-digraph 3-edge-connected, if its underlying graph is 3-edgeconnected. An even subgraph E of a digraph D = (V, A) is a subset E A that is an edge disjoint union of weak cycles of D. The main purpose of this section is to prove that Conjecture 1 is equivalent to the following: Conjecture 2. In every 3-edge-connected planar digraph D = (V, A) there exists an even subgraph E A such that D/E is strongly connected. Proof of equivalence of Conjectures 1 and 2. Assume Conjecture 1 holds, let D = (V, A) be a 3-edge-connected planar digraph and D = (F, A) its directed dual. Since D has no cut of size at most 2, D is an orientation of a simple graph. By Conjecture 1 and Theorem 1 D has a NL-2-coflow which becomes a NL-2-flow (f 1, f 2 ), when considered in D. Since f 1 is a flow using only values in {0, 1, 1}, its support must be the arc set E of an even subdigraph of D. Contracting E, the second component f 2 becomes a strictly positive integer vector in the cycle space of D/E, which can be decomposed into a sum of not necessarily disjoint directed cycles. Hence D/E is strongly connected. Now, assume Conjecture 2 holds and let D = (V, A) be an orientation of a simple planar graph. Then its directed dual graph D = (F, A) is 3-edgeconnected and hence has an even subgraph E such that D /E is strongly connected. The decomposition of E into weak cycles yields a flow f 1 in D with support E and values in {0, 1, 1}. A directed ear decomposition of D /E yields a strictly positive flow on A \ E which extends to a flow f 2 in D such that (f 1, f 2 ) are a NL-2-flow in D. Using duality this yields a NL-2-coflow in D and hence by Theorem 1 a 2-coloring of D. Tutte [17] conjectured that the Petersen graph is the only obstruction to the existence of a NZ-4-flow. Clearly this conjecture implies the 4-Color-Theorem. Suprisingly, life seems to become easier in the directed case. Proposition 2. Every orientation of the Petersen graph admits a NL-2-flow. Proof. It suffices to show that there always exist two vertex-disjoint 5-cycles, the complement of a perfect matching, such that the matching edges are not all oriented the same way with respect to the cycles. Starting with the pentagon and 4

7 Figure 1: Any orientation of the Petersen graph has a NL-2-flow the pentagram we are done, if the complementary matching edges are oriented not all the same way. Therefore, and by symmetry, we may assume that all edges are directed from the pentagram to the pentagon. Now considering the dotted circuits in Figure 1 and the complementary edge to the uppermost vertex we are done, if not all matching edges are oriented towards the upper cycle. Using the symmetry of the Petersen graph and rotating the configuration we find two cycles whose contraction leaves a strongly connected digraph. Since the Petersen graph is not an obstruction to the existence of a NL-2-flow we are tempted to conjecture: Conjecture 3. In every 3-edge connected digraph D = (G, A) there exists an even subgraph E A such that D/E is strongly connected. 4 NL-flows in 2,3 or 4-edge connected digraphs The following proposition shows that in a certain sense NL-flows are a proper generalization of NZ-flows. Given a bridgeless graph G = (V, E) its directed subdivision is the bipartite digraph S D (G) = (V E, Ã), where for each edge e = (u, v) E we have the two arcs (u, e) and (v, e) in Ã. Note, that S(G) arises from G by subdividing all of its edges and directing them towards the subdividing vertex. Proposition 3. A bridgeless graph G has a NZ-k-flow if and only if S(G) has a NL-k-flow. Proof. Assume S(G) has a NL-k-flow f = (f 1, f 2 ). Then for all e = (u, v) A the flow condition in e yields f((u, e)) = f((v, e)). Hence f 1 must be a nowhere-zero-k-flow of S(G). Both implications of the proposition now follow from the following bijection between NZ-k-flows f 1 of S(G) and NZ-k-flows (D, φ) of G, where D = (V, A). If a = (u, v) A is the orientation of edge e we have φ(a) = f(u, e), and for the orientation (v, u) on the other hand φ(a) = f(v, e). Then clearly φ satisfies 5

8 Kirchhoff s law if and only if f 1 does. Since f 1 is nowhere-zero, we may choose f 2 constant to zero. Thus, in general we have the same upper bound on the the minimum k for bridgeless digraphs in NL-flows as for NZ-flows for bridgeless graphs, namely 5 if Tutte s Five-Flow-Conjecture [16] is true, respectively 6 by Seymours 6-Flow- Theorem [15]. In order to show that we can do better, if the edge-connectivity is higher we need some preparations: Definition 4. We say that an integer flow φ in an orientation of a connected graph G is coindependent if its support contains a spanning tree of G. We say that G admits a coindependent k-flow, if some orientation (and thus any orientation) of G has a coindependent flow satisfying φ(e) < k for all e E(G). Proposition 5. If a connected graph G has a k-coindependent flow φ, then any orientation of G admits a NL-k-flow. Proof. We set f 1 = φ. Since supp(φ) contains a tree spanning G, for each e E(G) \ supp(φ) we find a C e such that e C e supp(φ) {e}. This yields a 2-flow φ e which is positive on e. Putting f 2 = we find a NL-k-flow (f 1, f 2 ). e E(G)\supp(φ) Let G = (V, E) be a graph and T V. A subset E 0 E of the edges is a T -join if and only if the set of vertices of odd degree in the subgraph G 0 of G induced by E 0 consists precisely of T. The following probably is folklore. We provide its proof for completeness. Proposition 6. A tree B = (V, E) posesses a T -join, if and only if T is even. Proof. By the classical Handshake Lemma the condition is necessary. We will prove sufficiency by induction on V, the case V = 1 being trivial. Hence assume n 2, T B V of even cardinality and let v be a leaf in B with neighbor w. If {v, w} T B we define T B\v := T B \ {v, w}, if v T B und w / T B we set T B\v := (T B \ {v}) {w}, and finally, if v / T then T B\v := T B. In any case T B\v is of the same parity as T B, and hence is even. By induction B \ {v} has a T B\v -join B 0. If v T B, by construction B 0 := B 0 {vw} is a T B -join of B; if v / T B, so is B 0 := B 0. The next tool we need is from the classical paper of Jaeger [10]. Theorem 7 (Kundu [11], 1974, Jaeger, 1979). (i) If G is a 4-edge-connected graph, then G has two edge-disjoint spanning trees B 1, B 2. (ii) If G is a 3-edge-connected graph, then G has three spanning trees B 1, B 2, B 3 such that no edge of G is contained in all three of them. φ e 6

9 Now we are ready to prove: Theorem 8. Let D be a directed graph with underlying graph G. (i) If G is 2-edge-connected, D admits a NL-6-flow. (ii) If G is 3-edge-connected, D admits a NL-4-flow. (iii) If G is 4-edge-connected, D admits a NL-2-flow. Proof. (i) This is immediate from Seymour s 6-flow Theorem[15] setting f 2 identically to zero. (ii) By Theorem 7, G has three spanning trees B 1, B 2, B 3 such that no edge of G is contained in all three of them. Let T i V denote the set of vertices of odd degree in B i. By the Handshake Lemma, T i is even for i = 1, 2, 3 and hence by Proposition 6, T i+1 contains a T i -join E i for i = 1, 2. Denote by D i the symmetric difference (E(T i ) E i ) \ (E(T i ) E i ). Then the sets D i induce even subdigraphs of D and hence there exist 2-flows φ 1, φ 2 with support D 1 resp. D 2. We define φ(e) = φ 1 (e) + 2φ 2 (e) and claim that this is a coindependent 4-flow of D. Clearly φ(e) 3 for all e E. Now let e B 1. If φ 1 (e) = 0 then e B 1 B 2 implying e B 3 and thus φ 2 (e) = 1. Hence the support of φ contains B 1, which proves the claim. Thus any orientation of G admitsa NL-4-flow by Proposition 5. (iii) By Theorem 7, G has two edge disjoint spanning trees B 1, B 2. Adding a T 1 -join in B 2 for the odd vertices of B 1 to the edge set of B 1 yields an even graph containing a spanning tree, and hence yields a coindependent 2-flow of D and the result follows. In (iii) of Theorem 8 we have proved the existence of a Eulerian subgraph which is spanning and connected. It was observed by Hakimi and Schmeichel [7] that in planar graphs this is dual to having vertex-arboricity 2, i.e. the vertex set can be partitioned into two classes each of them inducing a forest. This justifies the name coindependent: if the support of the flow contains a spanning tree, its complement is independent in the dual matroid. In the planar case (iii) becomes a result of Raspaud and Wang [14] that a triangle free planar graph is of vertex arboricity 2. It is known [2] that planar graphs have vertex arboricity at most 3 and hence have a coindependent 3-flow in the 3-edge-connected case (see also Proposition 15). Matt DeVos [3] brought to our attention that a result of Seymour [15] implies the existence of a coindependent 3-flow in the general 3-edge-connected case. We sketch the proofs for sake of completeness. Theorem 9 ([15]). Let G = (V, E) be a cubic 3-vertex-connected graph. Then G admits a coindependent 3-flow. Proof. Seymour defined the 2-closure X 2 of a set X E as the smallest set Y E containing X such that every circuit C of G either is contained in Y 7

10 or has at least 3 edges outside of X. Then he proved ([15], (5.1)) that every 3-vertex-connected cubic graph has a subset of edges X such that X 2 = E and E \ X contains a spanning tree. Now ([15], (3.1)) yields a 3-flow with E \ X in its support. The standard proof technique of flow theory reducing 3-edge-connected graphs to cubic 3-vertex-connected graphs implies. Theorem 10 (DeVos 2016, [3]). Let G be a 3-edge-connected (multi-)graph. Then G admits a coindependent 3-flow. Proof. Let G = (Ṽ, G) denote a 3-edge connected (multi)-graph such that G = G/X is a contraction minor for some edge set X Ẽ and (deg G(v) 3) v Ṽ is minimal. We claim that G must be cubic. For assume to the contrary that there exists a vertex v Ṽ such that deg G(v) 4. By a theorem of Fleischner [6] there exist e 1 = (v, u 1 ), e 2 = (v, u 2 ) Ẽ such that G \ {e 1, e 2 } {(u 1, u 2 )} is bridgeless and connected. Thus, if we define G by adding a vertex v which is adjacent to v, u 1 and u 2 to G and removing e 1, e 2, then G contradicts the minimality assumption for G. Hence G is cubic and 3-edge-connected, and thus also 3-vertex-connected. By Theorem 9 G has a coindependent 3-flow f. Now, f Ẽ\X is a coindependent 3-flow for G. Corollary 11. Let D be a directed graph with underlying 3-edge-connected graph G. Then D admits a NL-3-flow. All results in this section make no use of the particular orientation of a graph. The Petersen graph and also the planar examples of vertex arboricity 3 show that there cannot be any progress towards Neumann-Lara s 2-Color-Conjecture without taking the orientation into account. Unfortunately, we have no idea yet how to do so in general. 5 Oriented Matroids There is a natural way to generalize the above to oriented matroids in the same way as Tutte s coloring and flow theory for regular matroids was generalized to oriented matroids by Hochstättler, Nešetřil [8] and, later, by Hochstättler and Nickel [9]. Assume we are given an oriented matroid O on a finite set E represented by its covectors. By D we denote its set of cocircuits and for D D by D its signed characteristic function, i.e. 1 if e D + D(e) := 1 if e D 0 otherwise. 8

11 Recall that the chromatic number χ(o) of an oriented matroid is defined as the smallest k such that the lattice of coflows { } F (O) := λ DD λd Z (2) D D contains a coflow f F (O) such that 0 < f(e) < k for all e E. A NL-coflow in an oriented matroid is a tuple (f, f + ) F (O) O such that whenever f (e) = 0, for some element e E, then necessarily f + (e) = +. The dichromatic number χ(o) is defined as the smallest k, such that there exists a NL-coflow (f, f + ) such that f (e) < k for all e E. Note that replacing each element in an oriented matroid with a pair of antiparallel elements the dichromatic number of the constructed oriented matroid is the chromatic number of the original one. Hence, as in the graphic case the dichromatic number is a proper generalization of the chromatic number. Clearly the chromatic number of the underlying reorientation class of an oriented matroid is always an upper bound for the dichromatic number. Furthermore, the dichromatic number is 1 if and only if the oriented matroid is acyclic, meaning that O contains the all +-vector. Similarly to Definition 4 we call a coflow coindependent if the complement of its support is independent in the underlying matroid. The analogue to Proposition 5 now reads as follows. Proposition 12. If an oriented matroid O has a k-coindependent coflow φ, then any of its reorientations has dichromatic number at most k. Proof. Let f 1 be a k-coindependent coflow of (a reorientation of) O. Let B denote a basis of the matroid M underlying O containing the zero-set I = {e 1,..., e t } of φ. For all e I let C e denote the fundamental signed cocircuit of e with respect to B which is positive on e. Now we set and have an NL-k-coflow for O. f 2 = C e1... C et Hence the dichromatic number of an oriented matroid which has a cospanning cocircuit is bounded by 2. Corollary 13. Let O be a uniform oriented matroid of rank r 1 on n elements which has an independent hyperplane, i.e. a flat F in the underlying matroid such that rk(f ) = r 1 = F. Then χ(o) 2. Corollary 14. If O is a uniform oriented matroid, the orientation of a paving matroid or the cographic matroid of a Hamiltonian graph, then χ(o) 2. 9

12 Finally, we make the connection between coindependent flows and the vertex arboricity precise. Proposition 15. Let O denote an orientation of the graphic matroid M(G) of a graph G. Then O has a coindependent k-coflow if and only if the vertex set of G can be partitioned into k classes, each inducing a forest. Proof. If O has a coindependent k-coflow, then by Proposition 12 and its proof O, and hence also both orientations of G with oriented matroid O, have dichromatic number at most k and admit a NL-k-coflow (f 1, f 2 ) where f 1 is a coindependent coflow. The complement of the support of f 1 is independent and thus consists of the edges of a forest of G. Each of its connected components must be monochromatic. Clearly, we cannot have an edge of non-zero flow between two vertices of the same color. So, each color class induces a forest. On the other hand a partition of the vertices, where each class induces a forest, immediately yields a k-coloring and, as byproduct, a coindependent k-coflow. The chromatic number of an orientable matroid seems to be large only on orientations of graphic matroids. More precisely, in [9] Theorem 3 it is proven that the chromatic number of an oriented matroid is bounded by rk(o) + 1 and the bound is attained only for orientations of the polygon matroid of the complete graph K n+1. It might be similar with the dichromatic number. This in particular might apply to the following conjecture of Erdős and Neumann- Lara [5]: Conjecture 4 (Erdős and Neumann-Lara [5]). There is a function f : N N such that for every k N every graph with chromatic number f(k) has an orientation which has dichromatic number at least k. We conjecture that the same holds for oriented matroids even with the same function: Conjecture 5. If f : N N is a function satisfying the assertion from Conjecture 4, k N and O is an oriented matroid with χ(o) f(k), then there exists a reorientation I O of O such that χ( I O) k. References [1] A. Björner, M. L. Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented Matroids, Cambridge University Press, Mar [2] G. Chartrand and H. V. Kronk, The point-arboricity of planar graphs, Journal of the London Mathematical Society, s1-44 (1969), pp [3] M. DeVos. personal communication. [4] R. Diestel, Graph Theory, Springer, 3rd ed., Feb

13 [5] P. Erdős, Problems and results in number theory and graph theory. Numerical mathematics and computing, Proc. 9th Manitoba Conf., 1979, 3-21 (1980). [6] H. Fleischner, Eine gemeinsame Basis für die Theorie der Eulerschen Graphen und den Satz von Petersen, Monatshefte für Mathematik, 81 (1976), pp [7] S. L. Hakimi and E. F. Schmeichel, A note on the vertex arboricity of a graph, SIAM J. Discret. Math., 2 (1989), pp [8] W. Hochstättler and J. Nešetřil, Antisymmetric flows in matroids, Eur. J. Comb., 27 (2006), pp [9] W. Hochstättler and R. Nickel, On the chromatic number of an oriented matroid, J. Comb. Theory, Ser. B, 98 (2008), pp [10] F. Jaeger, Flows and generalized coloring theorems in graphs, Journal of Combinatorial Theory, Series B, 26 (1979), pp [11] S. Kundu, Bounds on the number of disjoint spanning trees, J. Combinatorial Theory Ser. B, 17 (1974), pp [12] V. Neumann-Lara, The dichromatic number of a digraph, Journal of Combinatorial Theory, Series B, 33 (1982), pp [13], Vertex colourings in digraphs. some problems., tech. rep., University of Waterloo, [14] A. Raspaud and W. Wang, On the vertex-arboricity of planar graphs, European Journal of Combinatorics, 29 (2008), pp Homomorphisms: Structure and Highlights. [15] P. D. Seymour, Nowhere-zero 6-flows, J. Combin. Theory Ser. B, 30 (1981), pp [16] W. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math., 6 (1954), pp [17], On the algebraic theory of graph colorings, Journal of Combinatorial Theory, 1 (1966), pp