Lehrstuhl für Mathematische Grundlagen der Informatik

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1 Lehrstuhl für Mathematische Grundlagen der Informatik Luis Goddyn, Petr Hliněný, W. Hochstättler: Balanced Signings of Oriented Matroids and Chromatic Number Technical Report btu-lsgdi Contact: Brandenburgische Technische Universität Cottbus Institut für Mathematik Lehrstuhl für Mathematische Grundlagen der Informatik Postfach D 033 Cottbus

2 2000 Mathematics Subject Classification: 05B35,52C40,05C15 Keywords: Oriented matroid, pseudosphere configuration, hyperplane arrangements,flow, circular chromatic number, discrepancy, cocircuit, wiring diagram

3 Balanced Signings of Oriented Matroids and Chromatic Number (final version May 7, 2003) Luis Goddyn * Department of Mathematics Simon Fraser University Burnaby, BC, Canada goddyn@math.sfu.ca and Petr Hliněný (corresponding author ) Institute of Theoretical Computer Science (ITI MFF) Charles University Prague, Czech Republic do not use for mail! hlineny@member.ams.org and Winfried Hochstättler Department of Mathematics BTU Cottbus, Cottbus, Germany hochst@math.tu-cottbus.de We consider the problem of reorienting an oriented matroid so that all its cocircuits are as balanced as possible in ratio. It is well known that any oriented matroid having no coloops has a totally cyclic reorientation, a reorientation in which every signed cocircuit B = {B +, B } satisfies B +, B. We show that, for some reorientation, every signed cocircuit satisfies 1/f(r) B + / B f(r), where f(r) 14 r 2 ln(r), and r is the rank of the oriented matroid. In geometry, this problem corresponds to bounding the discrepancies (in ratio) that occur among the Radon partitions of a dependent set of vectors. For graphs, this result corresponds to bounding the chromatic number of a connected graph by a function of its Betti number (corank) E V

4 2 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER 1. INTRODUCTION This paper regards an optimization problem which is an oriented matroid analogue of the graph chromatic number. There are several ways in which a chromatic number might be defined for more general matroids. One such formulation, introduced by Goddyn, Tarsi and Zhang [7], depends only on the sign patterns of (signed) circuits (or cocircuits). The result is a natural invariant of an oriented matroid. The invariant can be viewed as a discrepancy in ratio of a pseudo-hyperplane arrangement, and thus should be of interest to geometers. The main theorem answers a question raised in [7]. We first state the result and some consequences, using a minimal set of definitions. Detailed definitions appear in Section 2. It is convenient to view an oriented matroid O to be a matroid in which every circuit C (and cocircuit B) has been partitioned C = C + C, (and B = B + B ) subject to a standard orthogonality condition. We regard each bipartition as an unordered pair {C +, C }, where one of the parts may be empty. For I E(O), the reorientation O I of O is the new oriented matroid obtained from O by repartitioning each circuit C (and cocircuit B) according to the rules {C +, C } {C + (I C), C (I C)} and {B +, B } {B + (I B), B (I B)}. (1) where is the symmetric difference operator. Theorem 1.1. Let O be an oriented matroid of rank at most r. There exists a reorientation O I in which every cocircuit B of size at least two satisfies B +, B B /f(r) where f(3) 17, and f(r) 14r 2 lnr for r 3. For R-represented matroids, our result specializes to a new bound on the discrepancies in ratio that occur among the Radon partitions of minimally dependent sets of real vectors of small corank. To be more precise: Corollary 1.2. Let (v e : e E) be a list of nonzero vectors in R r. Then it is possible to replace some of these vectors by their negatives such * Supported in part by the National Sciences and Engineering Research Council of Canada, and the Pacific Institute for the Mathematical Sciences. Corresponding author s mailing address: Petr Hliněný, Institute of Mathematics and Comp. Science (MÚ SAV), Severná ulica 5, Banská Bystrica, Slovakia. Supported by Ministry of Education of the Czech Republic as project LN00A056.

5 BALANCED SIGNINGS AND CHROMATIC NUMBER 3 that, for any minimal sublist {v e : e C}, C E of linearly dependent vectors, every nontrivial real solution to e C α ev e = 0 has at least C /f( E r) coefficients α e of each sign, where f(s) 14s 2 lns. We may restate this result in the dual. The support of a vector t = (t e ) R E is supp(t) = {e E t e 0}. The rowspace of a matrix A is the set of vectors of the form ya, where y is a row vector. Corollary 1.3. Let A R R E be a real matrix of rank r, where R and E are index sets. Then it is possible to multiply some columns of A by 1 so that, for every vector t in the rowspace of the resulting matrix, if supp(t) 2 and supp(t) is minimal among the nonzero vectors in the rowspace, then t contains at least supp(t) /f(r) positive entries, where f(r) 14r 2 lnr. We remark that, if (v e : e E) are the columns of a real matrix A, then the sets C E, and supp(t) E referred to in Corollaries 1.2 and 1.3 are, respectively, the circuits and cocircuits of the oriented matroid represented by the matrix A. If, in Corollary 1.2, each vector v e is the difference of two unit vectors, then we are in the setting of graph theory. A formula of Minty [9] relates the graph chromatic number χ(g) to ratios of the form C / C + seen among reorientations of a graph G. Here our result further specializes to the observation that χ(g) f(rk (G)) for any loopless graph G = (V, E), where rk (G) = E V + 1 is the corank, or first Betti number of G. For graphs, the upper bound on f can be improved to χ(g) f(s) 2s. (This follows without much difficulty from Dirac s density result [3] for colour critical graphs.) This bound on χ(g) is (essentially) attained by complete graphs. The above discussion indicates that the best function f for our result satisfies 2s f(s) 14s 2 lns. We do not attempt to further optimize the function f(s) of Theorem BACKGROUND AND DEFINITIONS Here we give some motivating comments and make precise definitions. We are assuming basic familiarity with graphs and matroids, but we review some aspects of oriented matroids which may be less familiar to some readers. Among the several equivalent formulations of oriented matroid, the following definition, which is due to Bland and Las Vergnas [2] (cf. [1, Theorem 3.4.3]), is best suited to our needs. A signing of a set X is an unordered bipartition X = {X +, X } of X = X + X, where either part may be empty. We define the imbalance

6 4 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER or log-discrepancy of X to be imbal( X) = X min{ X +, X }, where the value is allowed if one of X +, X is empty. A pair ( C, B) of signed sets is orthogonal if (C + B + ) (C B ) = (C + B ) (C B + ) =. (2) This models the fact that a pair of vectors in Euclidean space is orthogonal only if either their supports are disjoint, or there is a positive and a negative summand in their scalar product. Any orientation G of a graph G naturally signs (the edge set of) each circuit C in G and of each cocircuit (minimal edge cut) B in G. Moreover, each signed circuit-cocircuit pair ( C, B) is orthogonal. Accordingly, we define an oriented matroid on the ground set E to be a triple O = (M, C, B) where 1. M is a matroid with ground set E, circuits C, and cocircuits B. 2. C = { C : C C} and B = { B : B B} are signings of the circuits and cocircuits such that each pair in C B is orthogonal. For any I E, the reorientation O I defined by (1) is easily seen to be an oriented matroid. This is analogous to reversing a subset I of the edges of a directed graph. Minty [9] considered the graph invariant χ o (G) := min G max C imbal( C), where G varies over the set of orientations of G, and C varies over the set of signed circuits in G. He showed that the graph chromatic number is given by χ(g) = χ o (G). The invariant χ o (G), now called the circular chromatic number, has several equivalent definitions and has seen a flurry of recent interest (see [15] for a survey). Within graph theory, this invariant is more usually denoted by χ c or χ, but we use χ o to emphasize the viewpoint of orientations. The definition of χ o is suited for generalization to oriented matroids. In the matroid setting, we prefer to speak in terms of the dual parameter. Thus we define the oriented flow number of an oriented matroid to be φ o (O) := min O I max B imbal( B),

7 BALANCED SIGNINGS AND CHROMATIC NUMBER 5 where O I varies over the set of reorientations of O, and B varies over the signed cocircuits in O I. Thus χ o (G) is just the cographic instance of φ o (O). In case O is graphic, φ o (O) = φ c (G) is the circular flow number of a graph G. This graph invariant, essentially introduced by Tutte [14], is also of contemporary interest [13, 16]. Seymour [12] showed that φ c (G) 6 for any 2-edge connected graph G. More generally if A is a totally unimodular matrix, then O = O[A] is an oriented regular matroid, and there is an algebraic description of the oriented flow number implicit in the work of Hoffman [8] and explicit in [7]. φ o (O[A]) = inf{α R x R E, Ax = 0, 1 x e α 1 for e E }. A coloop is a cocircuit of size one. If O has a coloop, then χ o (O) =. The converse also holds, since every coloop-free oriented matroid has a totally cyclic reorientation, one in which every signed cocircuit B satisfies B +, B. With this terminology, we may restate our main result. Theorem 2.1. For any coloop-free oriented matroid O of rank r, φ o (O) 14r 2 lnr. The dual of an oriented matroid O = (M, C, B) is the oriented matroid O = (M, B, C), where M is the dual matroid of M. Thus we may view the chromatic number of O to be the invariant χ o (O) = φ o (O ). Incidently, χ o (O) is much easier than φ o (O) to bound by a function of the rank r = rk(o). Since no circuit of O has cardinality more than r + 1, no circuit of O has imbalance greater than r + 1 in a totally cyclic orientation of O. It follows that χ o (O) r + 1 for any loopless oriented matroid O of rank r. This bound is best possible since it is achieved when O is (an orientation of) the complete graph M(K r+1 ). An orientation of a matroid M is an oriented matroid of the form (M, C, B). We say M is orientable if it has an orientation. Not all matroids are orientable. For example, a binary matroid is orientable if and only if it is regular. The orientations of M are partitioned into reorientation classes, where the class of O is [O] = {O I : I E}. For sake of brevity, we say that [O] is a reclass of M. An orientable matroid may have several distinct reclasses. However all regular matroids, including graphs and cographs, have a unique reclass. If O = (M, C, B) and I E(O) is a connected component of M, then O I = O. Indeed each reclass of M contains precisely 2 E ω orientations of M, where ω is the number of connected components of M. Matroidal notions such as independence, flats

8 6 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER and the rank function rk( ) carry over from M to any orientation O of M, and to any reclass [O]. The oriented flow number φ o ([O]) of a reclass is well defined. The oriented flow number of an orientable matroid M is not well defined. Example 2.1. The uniform matroid U 3,6 is orientable and has precisely four reclasses, [O i ], 1 i 4 (Figure 1). They satisfy φ o ([O 1 ]) = 2 and φ o ([O i ]) = 4 for i = 2, 3, 4. Any simple rank 3 reclass [O] may be conveniently viewed as a 2-pseudosphere complex which is affinely represented by a wiring diagram. A wiring diagram is a set of smooth plane curves {C e : e E}, each joining a pair of opposite boundary points of a fixed disc D in the plane. Any two curves must cross exactly once, and are otherwise disjoint. Each crossing point x is a vertex which corresponds to the matroid hyperplane {e E : C e x} and the complementary cocircuit {e E : C e x} B(O). An oriented matroid in [O] is determined by designating, for each e E, one of the two components of D C e as being the positive side of C e. Each element e of a cocircuit B is signed according to whether the corresponding vertex lies on the positive or negative side of C e. If O is not simple, then all elements of one parallel class [e] are associated to the same curve C e, although each element in [e] is independently oriented. We may record the cardinality of [e] on the diagram, and use arrows to indicate the positive side of each element. The boundary of D is called the equator of the diagram. Several wiring diagrams may correspond to the same oriented matroid since there is choice in the location of the equator O 1 O 2 O 3 O 4 FIG. 1. The four reclasses of the uniform matroid U 3,6. Figure 1 depicts the wiring diagrams of the four reclasses O 1, O 2, O 3, O 4 of U 3,6. We remark that all four diagrams could be drawn with straight lines. The first diagram O 1 has an orientation with imbalance 2. There exist no such orientations for the next three diagrams. (To see why, notice that the cocircuits have all size 4, and so imbalance must be either 2 or 4. Now consider one of the little triangles ; its lines have one orientation, up

9 BALANCED SIGNINGS AND CHROMATIC NUMBER 7 to symmetry, possibly achieving imbalance 2. This partial orientation then uniquely extends further, but cocircuits with imbalance 4 arise anyway.) The following additional facts may be found in [1], and are used tacitly in this paper. 1. An oriented matroid (M, C, B) is uniquely determined by either of the pairs (M, C) or (M, B). 2. For disjoint subsets S, T E(O), the minor O/S\T is the orientation of the matroid minor M/S\T obtained by restricting the signings in C B to the circuits and cocircuits of M/S\T. 3. There is a bijective correspondence between simple reclasses [O] (having no loops or multiple elements) and topological objects called pseudosphere complexes. This is due to Folkman and Lawrence [6] and Edmonds and Mandel [10]. 4. Any matroid representable over R is orientable. Every R-representation M = M[A], A R R E determines an orientation O = O[A] of M. The signings of the cocircuits are derived as follows: Consider the columns A.e of M as linear forms in R R. We may identify the 1-dimensional space spanned by that form in dual space with its nullspace and thus with a centrally symmetric hyperplane. If M[A] is simple, then the pseudo-sphere complex corresponding to [O[A]] is a complex S of subspheres of S := {x R R : x = 1}. Each matroid flat F E corresponds to the subsphere S F = S nullspace(a F ) S, where A F is the submatrix of A T whose rows are selected by F. The elements of M correspond to hyperspheres S {e} S. The hyperplanes of M correspond to 0-spheres in S. Each of the two points constituting a 0-sphere is called a vertex of S. The signing {B +, B } of a cocircuit B in O[A] is given via B + = {e B : A {e} x > 0} B = {e B : A {e} x < 0} where x is one of the two vertices corresponding to the hyperplane E B. 3. LOW RANK The odd cogirth of an oriented matroid O is the least odd integer 2k + 1 such that O has a cocircuit of cardinality 2k+1. We define the odd cogirth bound number of O to be the rational number ocb(o) = 2 + 1/k, where 2k + 1 is the odd cogirth of O. If O has no odd cocircuits then we define its odd cogirth to be and ocb(o) = 2. The imbalance of any

10 8 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER signed cocircuit of size 2k + 1 is at least (2k + 1)/k. Thus we have the following. Proposition 3.1. ocb(o). For any oriented matroid O we have φ o (O) This bound is generally quite weak. It fails already for the graphic matroid M(K 4 ), and for the other orientations in Example 2.1. However, the bound is exact for oriented matroids of low rank. Lemma 3.2. Every oriented matroid O of rank at most 2 satisfies φ o (O) = ocb(o). In particular, φ o (O) 3 if O is coloop-free. Proof. If rk(o) = 1, then O is an orientation of a single parallel class. Orienting this cocircuit as equitably as possible gives φ o (O) = ocb(o). If the rank equals two, then we may affinely represent O as a list of points P = (p e : e E) on the real number line. Let us label the elements with e 1,..., e n so that p e1... p em. For each p P there is a corresponding cocircuit B p = {e E p e p}. We sign B p according to B + p = {e i : p ei < p and i is odd} {e i : p ei > p and i is even}. It is easy to verify that this gives an orientation of O, and that every cocircuit B e satisfies B + e B e {0, ±1}. It follows that imbal(b e ) equals 2 if B e is even, and equals 2 + 1/k if B e = 2k + 1. Therefore φ o (O) = ocb(o). 4. RANDOM RESIGNINGS AND RANK 3 We shall make use the Chernoff bound from probability theory. Lemma 4.1. If X 1,..., X m = ±1 are i.i.d. random variables with probability 1/2, then for λ > 0 we have Prob( X i > λ ) m < 2e λ2 /2. i Here is some convenient terminology. Let A be a subset of a cocircuit B, and let B = {B +, B } be a signing of B. For s 2, we say that A is s-unbalanced in B if A min{ A B +, A B } > s.

11 BALANCED SIGNINGS AND CHROMATIC NUMBER 9 If B is a signed cocircuit in an oriented matroid O, and B is s-unbalanced in B, then we say that B is s-unbalanced in O. Therefore φ o (O) = min I E inf{s R : no cocircuit is s-unbalanced in O I}. A random resigning of a subset R E(O) is a reorientation O I where I is uniformly selected among the subsets of R. Lemma 4.2. Let A be a nonempty subset of a cocircuit B in O, and let R satisfy A R E. Let B be the signing of B in a random resigning of R. Then for s 2 the probability that A is s-unbalanced in B is less than ( ( 2 exp 1 2 ) ) 2 A. s 2 Proof. We define random variables {X e : e A} by { 1 if e B + X e = 1 if e B. Then A is s-unbalanced in B if and only if e A X e > (1 2 s ) A. Since A R, the random variables are i.i.d. among the resignings of R. The result follows by applying the Chernoff bound with λ = ( 1 s) 2 A. We shall now prove a bound on φ o (O) in case O has rank 3. Figure 2 shows three oriented wiring diagrams which are all simple, except for one element of multiplicity two or three. More precisely, each example is the case n = 4 of a family of wiring diagrams on n+3 elements. Orientations are given by the arrows shown. We say that these orientations are alternating with respect to the equators shown. The reader will notice that the orientation described in the proof of Lemma 3.2 is alternating in a similar sense. Alternating orientations of matroids of rank 3 tend to result in good upper bounds on φ o (O). Lemma 4.3. Let O be a coloop-free oriented matroid of rank 3. Then φ o (O) 17. Moreover, if O is simple and n = E(O) is large, then { ( φ o (O) max ocb(o), 2 + O (n/ lnn) 1/3)}. In particular, if n 159 (n 427), then φ o (O) 4 (resp. φ o (O) 3).

12 10 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER ¾ ½ ¾ Ò ¾ ½ ¾ ½ ¾ Ò Ò ½ ¾ ½ ¾ ½ ¾ FIG. 2. Three wiring diagrams with alternating orientations. Proof. We may assume O has no loops. Suppose that S = {f 1, f 2 } are parallel elements in O such that O\S is coloop-free. By orienting f 1 and f 2 oppositely and using induction, one easily sees φ o (O) φ o (O\S) 17. Thus we may assume that any two parallel elements of O are contained in a cocircuit of size 3. If O is not simple, then O has n+3 elements for some n 0, and O corresponds to one of the three wiring diagrams as drawn in Figure 2 (the cocircuit is {f 1, f 2, f 3 }). The alternating reorientations shown there, imply that φ o (O) 3 if O has no coloops. Thus we may assume O is simple. Let k = min{ B : B B(O)} be the cogirth of O. Let B 0 be a cocircuit of size k in O. We may choose the equator of a wiring diagram to be near the vertex corresponding to B 0, so that the left half of the disk is as shown in Figure 3 (the diagram is undetermined under the white box). We consider an alternating reorientation O of O as shown in the diagram. ¼ ¼ FIG. 3. An alternating reorientation O defined by a smallest cocircuit B 0. By design, we have imbal O (B 0 ) ocb(o) 3. Every other cocircuit B B(O ) {B 0 } contains all but possibly one element of E B 0. Since O is alternating, we therefore have B +, B (n k 1)/2. Since

13 BALANCED SIGNINGS AND CHROMATIC NUMBER 11 B n 2, this implies imbal O (B) g(n, k) where g(n, k) = n 2 (n k)/2 1 = 2n 4 n k 2. Thus the reorientation O shows that if 2 k n 3, then φ o (O) max {ocb(o), g(n, k) }. (3) We note that ocb(o) > g(n, k) only if k n 2. Let O I be a random resigning of E(O). Since O is simple, it clearly has at most ( n 2) cocircuits, all of which have size at least k. By Lemma 4.2, the probability that some cocircuit of O I is s-unbalanced is less than ( ) ( n 2 exp 2 ( 1 2 ) ) 2 k. (4) s 2 Let f(n, k) denote the least real number s > 2 such that this expression is less of equal 1. One can verify that f(n, k) is well defined for k > 2 ln ( 2 ( n 2)). When s f(n, k), at least one of the random reorientations O I has no s- unbalanced cocircuits. Therefore, if 4 lnn k n 2, then φ o (O) f(n, k). (5) For fixed n 5, we find that kg(n, k) > 0 for 2 k n 3, whereas kf(n, k) < 0 for 4 lnn k n 2. The bounds in (3) and (5) are illustrated in Figure 4 for n = 30. We note that ocb(o) is bounded above by 2k/(k 1), but is not determined when k is even. Using a computer algebra system, we find that g(n, k) = f(n, k) when k = k 0 = 3 2(n 2) 2 ln (n(n 1)). We have shown φ o (O) max{ocb(o), h(n)} where h(n) = g(n, k 0 ) = ( 2n 4 ( n ) ) 1/3 n 2 3 2(n 2) 2 ln(n(n 1)) = 2 + O. lnn Using the trivial bound φ o (O) max{ B B B(O)} n 2 and the easily verified fact that f(n, k 0 ) 17 when n 19, we have proven φ o (O) 17. We further find that φ o (O) 4 (resp. 3) provided that n 166 (resp. 712). However, we can improve the above argument in case we are trying to prove φ o (O) s for some s 4. When n 162 one finds

14 12 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER Ò ¼ Ò µ Ò µ Ô Ò ¾ ÐÒ Ò ¼ Ó Çµ k FIG. 4. f(n, k), g(n, k), and ocb(o) versus k, when n = 30. that k 0 < (n 1)/2, which roughly corresponds, by (3), to s 4. Since O is simple, it contains at most one cocircuit of size less than (n 1)/2, and at most ( ) n 2 ( ) n k 2 3n(n 2) 8 cocircuits of size at least (n 1)/2. For s ocb(o), we may assume k (n 2)(s 2)/s since otherwise φ o (O) s by (3). Therefore, in case k n/2 1, we may replace (4) with the smaller quantity ( ( 2 exp 1 2 ) ) 2 ( ) ( ( (n 2)(s 2) 3n(n 2) + 2 exp 1 2 ) ) 2 n 1. s 2s 8 s 4 It is straight forward to verify that this expression is at most 1 when n 427 (for s = 3), and when n 159 (for s = 4). With extra work, the minor restriction that O is simple can be omitted from the statement of Lemma 4.3. The bound φ o (O) 17 is probably far from optimal. Conjecture 4.4. Every coloop-free oriented matroid O of rank 3 satisfies φ o (O) DENSE FLATS There are additional challenges when considering oriented matroids of rank greater than three. It is not clear how to define an alternating

15 BALANCED SIGNINGS AND CHROMATIC NUMBER 13 orientation, so we shall resort to using random orientations. However a matroid of rank 4 may have have many small cocircuits, so we can not argue as in the proof of Lemma 4.3 to bound the probability of having an unbalanced cocircuit. We give here a way to address this problem. The following example helps illustrate the forthcoming strategy. Let G 1 be the complete graph K 10. For i 1, let G i+1 be the graph obtained from G i by replacing each vertex v V (G i ) with a copy K(v) of K 10. For every edge e E(G i ) joining u to v, there is an edge in G i+1, which is also labeled e, and which joins an arbitrary vertex of K(u) to an arbitrary vertex of K(v). Therefore V (G i ) = 10 i and G i = G i+1 /F i where F i = {E(K(v)) v V (G i )} (6) We observe that F i is a flat (closed set) of the matroid M(G i+1 ). Because the graph G 20 has small cogirth (9) and a large number of cocircuits (about ), a naïve application of Lemma 4.2 would result in a very poor bound on φ o (M(G 20 )). However, we observe that any cocircuit in G 20 which is small either has the majority of its edges in E(K(v)) for some v G 19 or is disjoint from F 19. With this observation, one can derive a much better bound on the imbalance of a random orientation of M(G 20 ). To apply this type of argument to an arbitrary oriented matroid O, we must first define a suitable set of elements in O which can play the role of F i in equation (6). Let O be an oriented matroid with E = E(O). A flat F is dense in O if F E rk(f) + 1 rk(e) + 1. A dense flat F is minimal if no proper subflat of F is dense in O. Since E is dense and is not dense, O has a minimal dense flat and all minimal dense flats are nonempty. A cocircuit B is F-intersecting if B F. Let B F denote the set of F-intersecting cocircuits in O. We first show that a substantial portion of any F-intersecting cocircuit lies within F. Lemma 5.1. Let F be a minimal dense flat in O, and let B B F. Then B F > E rk(e) + 1. Proof. Suppose not. Since B F, we have that F := F (E B) is a proper subflat of F in O satisfying F E = F E B F rk(f) + 1 E rk(e) rk(e) + 1 rk(f ) + 1 rk(e) + 1.

16 14 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER This contradicts that F is minimally dense in O. Lemma 5.2. Let O be an oriented matroid of rank r 3 and size n. Let R E(O) and F R be a minimal dense flat in O. Suppose that t R satisfies 19r 2 lnr t n. Then the probability that some F-intersecting cocircuit is t-unbalanced in a random resigning of R is less than n 2. If 3 is replaced by 4 or 5, then 19 may be replaced by 14 or 12 respectively. Proof. Let B B F. In any reorientation of O, we have by Lemma 5.1, B B + n B + B F B + F. where = r + 1. Thus in a random resigning of R, the probability that B is not t-balanced is at most the probability that B F is not t -balanced in B. By Lemmas 4.2 and 5.1 this probability is at most 2 exp ( We aim to show ( 1 2 ) 2 B F t 2 ) 2 exp ( ( 1 2 ) 2 n t 2 ) =: P. n 2 B F P < 1. (7) We estimate B F ( ) n r 1 n 2 2 (for r 3). Thus (7) holds provided that n P 2 < 1. Equivalently, we aim to show ( lnn 1 2 ) 2 n t 2 < 0, 2 lnn ( 1 n < 2 ) 2. (8) t For any integer r 0 3, let f(r 0 ) be the smallest positive number such that every integer r r 0 satisfies f(r 0 ) r 2 lnr e 1 f(r 0) r r 2(r 0 +1) 2. It is straight forward to verify that f(3) 19, f(4) 14, f(5) 12, and that f(r) 4 slowly. Writing t 0 = f(r 0 )r 2 lnr, and = r + 1, we have for r r 0 3, lnt 0 f(r 0) r 2 0 2(r 0 + 1) 2 lnr 1 2 f(r 0) r 2 2(r + 1) 2 lnr 2 = t

17 BALANCED SIGNINGS AND CHROMATIC NUMBER 15 Together with t 0 t n, this completes the proof since the left hand side ln t0 of (8) is at most 2 t 0, whereas the right hand side is at least t PROOF OF MAIN THEOREM Let O be an oriented matroid. To prove the bound on φ o (O), we shall construct an ordered partition (F 0, F 1,..., F p ) of E(O) as follows. Let F 0 be a minimal dense flat in O 0 = O and suppose we have constructed F 0, F 1,...,F k, for some k 0. If k i=0 F i = E(O), then we set p = k and output the list. Otherwise let F k+1 be a minimal dense flat in the contracted oriented matroid /( k ) O k+1 := O F i = O k /F k. i=0 Since each F k is nonempty and O is finite, this procedure must terminate. Any sequence (F 0, F 1,..., F p ) constructed in this way is called a dense flat sequence of O. Let (F 0, F 1,...,F p ) be a dense flat sequence for O. A cocircuit B of O is of type k if k is the least index for which B F k. Thus the cocircuits in O k are precisely the cocircuits of type k in O. Let B k be the set of cocircuits of type k in O, let n k = O k and let r k = r(o k ) = r(o) k 1 i=0 r(f i). Figure 5 may help the reader with these definitions and the proof that follows. ¼ ½ ¾ Ç ¾ Õ ½ ßÞ Ð ¾ ¾ Ç Õ Ö Õ ÓÖ Ò Õ Ø ¼ FIG. 5. A dense flat sequence and a cocircuit B of type 2 Theorem 6.1. coloops, For any oriented matroid O of rank r and without φ o (O) < 14r 2 lnr.

18 16 L. GODDYN, P. HLINĚNÝ, W. HOCHSTÄTTLER Proof. We show that O has a reorientation in which no cocircuit is t 0 -unbalanced, where t 0 = 14r 2 lnr. Let (F 0, F 1,...,F p ) be a dense flat sequence for O, with O k, B k, n k, r k as defined as above. Let q be the least integer for which either r q 3 or n q t 0. If r q 3, then by Lemma 4.3, O q may be reoriented so that every cocircuit in O q has imbalance at most 17. If n q t 0, then in any totally cyclic orientation of O q, we have that every cocircuit has imbalance at most t 0 2. Thus O has a reorientation O in which no cocircuit of type at least q is max{17, t 0 2}-unbalanced. We now randomly resign q 1 i=0 F i in O to obtain O. We aim to show that with probability greater than zero, no cocircuit is t 0 -unbalanced in O. By choice of q we have r k 4 and n k > t 0, for 0 k q 1. So applying Lemma 5.2 to O k and F k with R = q 1 i=k F i to find that that the probability that a cocircuit in B k is t 0 -unbalanced in O k is at most n 2 k. By definition each cocircuit from B k is contained in E(O k ). Since ( B 0, B 1,..., B q 1, B(O q ) ) is a partition of the cocircuits of O, the probability that some cocircuit in O is max{t 0, 17}-unbalanced is at most q 1 k=0 n 2 k Thus φ o (O) max{t 0, 17} = t 0. i 2 < 1. i=2 7. SMALL EXAMPLES Using an implementation by Lucas Finschi [4] of a simple reclass generator described in [5], together with a program of Timothy Mott [11], for calculating χ o (O), we have found the following results. For various values of n and s we list in Table 1 the number of coloopfree cosimple reclasses [O] of rank 3 and size n for which φ o ([O]) = s. In parentheses, we record the number of these which are reclasses of the uniform matroid U n,3. If O is not cosimple, we may contract any coparallel element to obtain a rank 2 oriented matroid with the same oriented flow number, which equals ocb(o) by Lemma 3.2. For rank 4, we find that, there are 143 reclasses [O] of size 7. The number of these which have φ o ([O]) = s is tabulated as follows, with parentheses indicating those which are uniform. s = 2 : 1(1) s = 3 : 12(0) s = 4 : 130(10). A natural question is whether the lower bound given by the Betti number of complete graphs is the worst case or not. We consider this question rather

19 BALANCED SIGNINGS AND CHROMATIC NUMBER 17 TABLE 1. Number of cosimple reclasses with rank 3, size n and oriented flow number s. In parentheses are listed the number of these which are uniform. s\n (0) 1 (1) 1 ( 0) 18 ( 3) 5/2 0 (0) 0 (0) 137 (11) 631 ( 0) 3 1 (1) 7 (0) 57 ( 0) 5369 (132) 4 0 (0) 9 (3) 11 ( 0) 11 ( 0) Total 1 (1) 17 (4) 206 (11) 6029 (135) hard, and so we make no conjectures here. However, we do not know of any construction of oriented matroids which would show that φ o (O) > Θ( rk(o)). Acknowledgment We thank the Pacific Institute of Mathematics for supporting the Thematic Programme on Flows, Cycles and Orientations, Burnaby, 2000, where much of this work was done. We also thank Lukas Finschi for making available for us a program for generating all signed circuits of all simple reclasses, and Timothy Mott for his program for calculating χ o (O). REFERENCES 1. A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler, Oriented Matroids, Encyclopedia of Mathematics and its Applications, 46, Cambridge University Press, Cambridge, R. G. Bland, M. Las Vergnas, Orientability of matroids, J. Combin. Theory Ser. B, 24 (1978), G. A. Dirac, The number of edges in critical graphs. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II. J. Reine Angew. Math. 268/269 (1974), L. Finschi, Catalog of oriented matroids, 5. L. Finschi, K. Fukuda, Generation of oriented matroids A graph theoretical approach, Discrete Comput. Geom. 27 (2002), J. Folkman and J. Lawrence, Oriented Matroids, J. Combin. Theory Ser. B, 25 (1978), L.A. Goddyn, M. Tarsi, C.-Q. Zhang, On (k, d)-colorings and fractional nowhere zero flows, J. Graph Theory, 28 (1998), A. J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, Combinatorial Analysis: Proc. Sympos. Appl. Math., Vol.10, R. Bellman and M. Hall Jr. Eds, American Mathematical Society (1960), pp G. J. Minty, A theorem on n-coloring the points of a linear graph, Amer. Math. Monthly 69 (1962)

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