On zeros of the characteristic polynomial of matroids of bounded tree-width

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1 arxiv: v1 [math.co] 7 Mar 2017 On zeros of the characteristic polynomial of matroids of bounded tree-width Carolyn Chun 1, Rhiannon Hall 2, Criel Merino 3 and Steven Noble 4 1 Mathematics Department, United States Naval Academy, Chauvenet Hall, 572C Holloway Road, Annapolis, Maryland , United States of America 2 Department of Mathematical Sciences, Brunel University London, Uxbridge, UB8 3PH, United Kingdom 3 Instituto de Matemáticas, Universidad Nacional Autónoma de México, México City, México 4 Department of Mathematics, Economics and Statistics, Birkbeck, University of London, Malet Street, London, WC1E 7HX, United Kingdom March 8, 2017 Abstract We prove that, for any prime power q and constant k, the characteristic polynomial of any loopless, GF (q)-representable matroid with tree-width k has no real zero greater than q k 1. chun@usna.edu rhiannon.hall@brunel.ac.uk merino@matem.unam.mx. Investigación realizada gracias al Programa UNAM- DGAPA-PAPIIT IN s.noble@bbk.ac.uk 1

2 1 Introduction For a graph G, the chromatic polynomial χ G (λ) is an invariant which counts the number of proper colourings of G when evaluated at a non-negative integer λ. However, the chromatic polynomial has an additional interpretation as the zero-temperature antiferromagnetic Potts model of statistical mechanics. This has motivated research into the zeros of the chromatic polynomial by theoretical physicists as well as mathematicians. Traditionally, the focus from a graph theory perspective has been the positive integer roots, which correspond to the graph not being properly colorable with λ colors. A growing body of work has begun to emerge in recent years more concerned with the behaviour of real or complex roots of the chromatic polynomial. Sokal [15] proved that the set of roots of chromatic polynomials is dense in the complex plane. In contrast, many other results show that certain regions are free from zeros. For planar graphs, the Birkhoff Lewis theorem states that the interval [5, ) is free from zeros. For more results along these lines, we direct the reader to the work of Borgs [1], Jackson [8], Sokal [14], Thomassen [16] and Woodall [17]. Perhaps one of the outstanding open questions concerning real zeros is to determine tight bounds on the largest real zero of the chromatic polynomial. One such bound is given in [14] and depends on the maximum vertex degree. For recent surveys see [13] and [4]. In matroids, the corresponding invariant is the characteristic polynomial. The characteristic polynomial of a loopless matroid M, with ground set E and rank function r, is defined as χ M (λ) = F L µ M (, F )λ r(e) r(f ), where L denotes the lattice of flats of M and µ M the Möbius function of L. When M has a loop, χ M (λ) is defined to be zero. Observe that a loopless matroid M and its simplification have the same characteristic polynomial. From this definition we see that, when M is loopless, χ M (λ) is monic of degree r(e). For example, for prime power q, the projective geometry P G(r 1, q), whose lattice of flats is isomorphic to the lattice of subspaces of the r-dimensional vector space over GF (q), has characteristic polynomial χ P G(r 1,q) (λ) = (λ 1)(λ q)(λ q 2 ) (λ q r 1 ). The largest root of the characteristic polynomial for a projective geometry is 2

3 therefore q r 1. The characteristic polynomial of the uniform matroid, U r,n, is χ Ur,n (λ) = r 1 k=0 ( 1)k( n k) (λ r k 1). For more background on matroid theory, we suggest that the reader consults [11]. For the theory of the Möbius function and the characteristic polynomial, we recommend [3, 18]. Perhaps the most compelling open question concerning real zeros in this context is deciding whether there is an upper bound for the real roots of the characteristic polynomial of any matroid belonging to a specified minorclosed class. Welsh conjectured that no cographic matroid has a characteristic polynomial with a root in (4, ). This was recently disproved by Haggard et al. in [6], and, in [9], Jacobsen and Salas showed that there are cographic matroids whose characteristic polynomials have roots exceeding five. Consequently, determining whether an upper bound exists for the roots of the characteristic polynomials of cographic matroids remains open. In [13], Royle conjectured that, for any minor closed class of GF (q)-representable matroids, not including all graphs, there is a bound on the largest real root of the characteristic polynomial. Furthermore, he suggested that the restriction to GF (q)-representable matroids might even be unnecessary. Given the situation with cographic matroids, this is clearly a difficult conjecture to resolve in the affirmative. In contrast, the situation with graphic matroids has been resolved. Thomassen [16] noted that, by combining a result that he and Woodall [17] had obtained independently with a result of Mader [10], one obtains the following. Theorem 1.1. Let F be a proper minor-closed family of graphs. Then there exists c R such that the chromatic polynomial of any loopless graph G in F has no root larger than c. For certain minor-closed families of graphs, one can find the best possible constant c. One such example is the class of graphs with bounded tree-width, a concept originally introduced by Robertson and Seymour [12]. A treedecomposition of a graph G comprises a tree T and a collection {X t } t V (T ) of subsets of V (G) satisfying the following properties. 1. For every edge uv of G, there is a vertex t of T such that {u, v} X t. 2. If p and r are distinct vertices in T, the vertex v is in X p X r, and q lies on the path from p to r in T, then v X q. 3

4 The width of a tree-decomposition is max t V (T ) X t 1 and the tree-width of a graph is the minimum width of all of its tree-decompositions. As its name suggests, graph tree-width measures how closely a graph resembles a tree. Matroid tree-width, which we will define later, measures how closely a matroid resembles a tree. If a graph can be obtained by gluing small graphs together in a tree-like structure, then it has small tree-width. Likewise, if a matroid can be obtained by gluing small matroids together along a tree-like pattern, then it has small matroid tree-width. Thomassen [16] proved the following. Theorem 1.2. For positive integer k, let G be a graph with tree-width at most k. Then the chromatic polynomial, χ G (λ), is identically zero or else χ G (λ) > 0 for all λ > k. Thomassen s proof proceeded essentially as follows, using induction on the number of vertices of G. Let G have tree-width k. Take a tree-decomposition of width k, with notation as above. Choose s and t to be neighbouring vertices in T. Then X s X t is a vertex-cut of G. One may add edges to G with both end-vertices in X s X t until X s X t forms a clique without altering the tree-width. Call this new graph G. The chromatic polynomial of G may be written in terms of the chromatic polynomial of graphs with fewer vertices than G having tree-width at most k and the chromatic polynomial of G in such a way that one may apply induction provided the result can be established for G. But since G has a clique whose vertices comprise a vertexcut, the chromatic polynomial of G may also be expressed in terms of the chromatic polynomials of graphs with fewer vertices and having tree-width at most k. In this paper we make some progress towards Royle s conjecture by extending the argument above using the matroidal analogue of tree-width, which we shall refer to as matroid tree-width, or simply tree-width, when the context is clear. This analogue of graph tree-width was developed by Hlilĕný and Whittle in [7]. A tree-decomposition of a matroid M is a pair (T, τ), where T is a tree and τ : E(M) V (T ) is an arbitrary mapping. For convenience, let V (T ) = {v 1, v 2,..., v l } and let E i = τ 1 (v i ) for all i in {1, 2,..., l}. We say that E i is the bag corresponding to v i. Let c i be the number of components in T v i, and let T i,1, T i,2,..., T i,ci denote the components in T v i. For j {1, 2,..., c i }, let B i,j be the subset of E(M) given by {e τ(e) V (T i,j )}. The vertex v i is said to display the partition B i,1, B i,2,..., B i,ci of E(M) E i. We say that the 4

5 T 5,1 B 5,1 T 5,2 B 5,2 v 1 v 2 E 1 E 2 v 5 E 5 v 6 E 6 T 5,3 T 5,4 v 4 E 4 v 3 E 3 B 5,4 B 5,3 v 8 E 8 v 7 E 7 Figure 1: A sample tree decomposition of a matroid M where E(M) = i {1,2,...,8} E i. rank defect of B i,j, denoted rd(b i,j ), is equal to r(m) r(e(m) B i,j ). Note that this number is the same as the size of the smallest set I B i,j such that all of the elements in B i,j I are in the closure of E(M) B i,j in the matroid M/I. Clearly I is an independent set in M. The rank defect is therefore a measure of the amount of rank contributed to M solely by the set B i,j. The node width of a vertex v i, written nw(v i ), is equal to r(m) c i rd(b i,j ). The width of (T, τ) is the maximum node width of all vertices in V (T ). The matroid tree-width of M, written tw(m), is equal to the minimum width of all tree-decompositions of M. It follows immediately from the definitions, that in any tree-decomposition the rank of each bag is at most the width of the tree-decomposition. We let v(m) be the number of vertices in the smallest tree over all of the tree-decompositions with width equal to the tree-width of M. For an example of a tree-decomposition, see Figure 1. The vertex v 5 displays the partition B 5,1, B 5,2, B 5,3, B 5,4 of E(M) E 5 indicated by the dashed regions. In this example, B 5,1 and B 5,2 each consist of one bag, whereas B 5,3 is the union of three bags, namely E 3, E 6, and E 7. The set B 5,4 is E 4 E 8. Each dashed region also indicates a subtree of T. For example, j=1 5

6 T 5,3 consists of the vertex set {v 3, v 6, v 7 } and edge set {v 3 v 6, v 3 v 7 }. In addition to that above, we employ an alternate use of the term display as follows. Let e = uw be an edge of T, let T u and T w be the two components of T \e containing u and w respectively, and let U and W be the sets of matroid elements U = {x τ(x) V (T u )} and W = {x τ(x) V (T w )}. We say that the edge e displays the sets U and W. The next result follows from Lemma 3.1, which we prove in Section 3. Lemma 1.3. Let M be a matroid with tree-width k. If (T, τ) is a treedecomposition of M of width k such that V (T ) = v(m), then, for every pair of subsets U and V of E(M) displayed by an edge of T, neither r(u) nor r(v ) is equal to r(m). Note that the tree-width of a matroid is at least equal to the tree-width of each of its minors, so the class of matroids with tree-width at most k is closed under taking minors. The following is the main result of this paper. Theorem 1.4. For prime power q and positive integer k, let M be a GF (q)- representable matroid with tree-width at most k. Then χ M (λ) is identically zero or else χ M (λ) > 0 for all λ > q k 1. It may seem restrictive to require that M be representable over a finite field, but every matroid that is representable over an infinite field is also representable over some finite fields (see, for example, [11, Corollary ]), so this restriction is relatively superficial. The requirement of representability, however, is essential to the result. For instance, the characteristic polynomial of the n-point line, U 2,n, has a root at n 1, hence a minor-closed class of matroids with bounded treewidth need not have an upper bound for the roots of their characteristic polynomials. The projective geometry P G(k 1, q) has tree-width k and its characteristic polynomial has a root at q k 1, hence the bound given is the best possible. 2 The characteristic polynomial The characteristic polynomial satisfies many identities similar to those satisfied by the chromatic polynomial. The following is one such identity, which is particularly important for us. 6

7 Theorem 2.1. If e is an element of a matroid M that is neither a loop nor a coloop, then the characteristic polynomial of M satisfies χ M (λ) = χ M\e (λ) χ M/e (λ). From Theorem 2.1, it is easy to see that a loopless matroid and its simplification have the same characteristic polynomial. The second identity which we will need is a special case of a result of Brylawski [2]. We first define the generalized parallel connection of two matroids M 1 and M 2 with ground sets E 1 and E 2, respectively, according to [11, page 441]. Let T = E 1 E 2 and suppose that M 1 T = M 2 T. Furthermore, suppose that cl M1 (T ) is a modular flat of M 1 and that each element of cl M1 (T ) \ T is either a loop or parallel to an element of T. Let N denote the common restriction M 1 T = M 2 T. Then the generalized parallel connection across N is the matroid P N (M 1, M 2 ) whose flats are precisely the subsets F of E 1 E 2 such that F E 1 is a flat of M 1 and F E 2 is a flat of M 2. Suppose a graph G has vertex set V and edge set E, where G = (V, E) = (V 1 V 2, E 1 E 2 ), such that G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are themselves graphs. It is a well-known result that, if the graph G = (V 1 V 2, E 1 E 2 ) is isomorphic to K k, the complete graph on k vertices, then the chromatic polynomial P G (λ) is equal to P G 1 (λ)p G2 (λ) P Kk. We now state Brylawski s result (λ) which generalizes this result to matroids. Theorem 2.2 (Brylawski (1975)). Let M be a generalized parallel connection of the matroids M 1 and M 2 across the modular flat N. Then χ M (λ) = χ M 1 (λ)χ M2 (λ). χ N (λ) 3 Bounds for zeros of the characteristic polynomial We now prove the following lemma, from which Lemma 1.3 immediately follows. Lemma 3.1. Let (T, τ) be a tree-decomposition of a matroid M. Suppose that T has an edge e = uw that displays the sets U, W E(M), where U cl(w ). Then there exists another tree-decomposition (T, τ ) of M having width at most the width of (T, τ), such that V (T ) < V (T ). 7

8 Proof. Consider T. Let T 1, T 2,..., T l be the connected components of T \w, where u T 1. Note that U = τ 1 (V (T 1 )). Let T be T \T 1. We define τ such that τ (x) = τ(x) if x / U, and τ (x) = w if x U. Clearly, V (T ) < V (T ). Take s V (T ). We will show that nw (T,τ )(s) = nw (T,τ) (s), and the result will follow. Let E s and E s be the bags corresponding to s in (T, τ) and (T, τ ), respectively, where s w. Then E s = E s, and the partition of E(M) E s displayed by s in (T, τ ) is the same as the partition of E(M) E s displayed by s in (T, τ). It follows that nw (T,τ )(s) = nw (T,τ) (s). We conclude this proof by showing that nw (T,τ )(w) = nw (T,τ) (w). In the original tree-decomposition, (T, τ), the set E w is the bag corresponding to w, and w displays the partition of the elements of E(M) E w into the sets B 1, B 2,..., B l, where B i = τ 1 (V (T i )). Note that B 1 = U. Then nw (T,τ) (w) = r(m) l rd(b i ). i=1 Similarly, in the new tree-decomposition, (T, τ ), the bag E w = E w B 1 corresponds to the vertex w, and w displays the partition of the elements of E(M) E w into the sets B 2, B 3,..., B l. It follows that nw (T,τ )(w) = r(m) l rd(b i ). i=2 However, since B 1 = U cl(e(m) U), we have rd(b 1 ) = 0. It follows that nw (T,τ )(w) = nw (T,τ) (w), as required. During the remainder of this section, for a simple GF (q)-representable matroid M, we denote by M q the projective geometry P G(r(M) 1, q) of which M is a spanning restriction. We also need a few definitions before proceeding to prove the main theorem. Take (T, τ), a tree-decomposition of M. For edge uw in T, let U and W be the subsets of E(M) displayed by uw, where τ 1 (u) U. Let U be a subset of elements of M q obtained by taking the closure cl M q (U ), and likewise, let W = cl M q (W ). We say that the neck of uw with respect to M q, or simply the neck of uw when the projective geometry is clear, is the set of elements in U W. Note that the neck of each edge is a projective geometry over GF (q). 8

9 Suppose there exists an element e in the neck of edge uw such that e / E(M). Consider the restriction of M q to the elements E(M) e. We denote this matroid by M e. Now M e has a tree-decomposition (T, τ ) obtained from (T, τ) by letting τ (x) = τ(x) when x E(M) and by letting τ (e) be either u or w. Thus the decomposition is the same except that we add e to the bag corresponding to u or the bag corresponding to w. We show that, for each edge of T, the corresponding subsets of E(M) and E(M e ) displayed by this edge have the same rank defects, and conclude that M and M e have the same tree-width. By the definition of rank defect, if e was added to a set B, then rd M e(b e) = r(m e ) r M e(e(m) B) = r(m) r M (E(M) B) = rd M (B). Hence the rank defect of B in M is equal to the rank defect of B e in M e. If e was not added to a set B that is displayed by an edge of T, then e is in the closure of E(M) B in M e by construction. The rank defect again remains unchanged, as rd M e(b) = r(m e ) r M e((e(m) B) e) = r(m) r M (E(M) B) = rd M (B). Therefore each vertex in T has the same node width in (T, τ) and (T, τ ). It follows that tw(m e ) = tw(m). Note that (M e ) f = (M f ) e if f is in the neck of uw but not in E(M) {e}, and we may unambiguously refer to this matroid as M {e,f}. For a set S of elements in the neck of uw, where S avoids E(M), the matroid M S is therefore well-defined. We now prove the main result of this paper. Proof of Theorem 1.4. If M has a loop, then its characteristic polynomial is identically zero, so we may assume that M is loopless. As M and its simplification have the same characteristic polynomial and the same treewidth, we may assume that M is simple. Suppose that r(m) = 1. Then M = U 1,1, and χ M (λ) = λ 1. Thus χ M (λ) > 0 if λ > 1, hence χ M (λ) is certainly positive for all λ > q k 1. We now assume r(m) > 1 and proceed by induction on r(m). Suppose that M has the smallest rank of all simple GF (q)-representable matroids with tree-width at most k such that χ M (λ) is not positive for all λ > q k 1. Take (T, τ), a tree-decomposition of M, where T has width equal to tw(m) and V (T ) = v(m). Suppose first that T contains a leaf edge uw, where w has degree one. Let S = {s 1, s 2,..., s n } be the elements in the neck of uw that are not in E(M). By Theorem 2.1, χ M (λ) = χ M s 1 /s1 (λ) + χ M s 1 (λ). By repeated application of 9

10 Theorem 2.1, χ M (λ) = χ M s 1 /s1 (λ) + χ M s 1 (λ) = χ M s 1 /s1 (λ) + χ M {s 1,s 2 } /s 2 (λ) + χ M {s 1,s 2 }(λ) = χ M s 1 /s1 (λ) + χ M {s 1,s 2 } /s 2 (λ) + χ M {s 1,s 2,s 3 } /s 3 (λ) + χ M {s 1,s 2,s 3 }(λ). = χ M s 1 /s1 (λ) + χ M {s 1,s 2 } /s 2 (λ) + + χ M S /s n (λ) + χ M S(λ). Since M and, consequently, M S are simple, contracting an element of S in M S results in a matroid that has rank equal to r(m) 1. Furthermore tw(m S ) = tw(m). Tree-width is not increased by contracting elements, so, by induction, the characteristic polynomial of the simplification of each matroid in {M s 1 /s 1, M {s 1,s 2 } /s 2,..., M S /s n } is positive for all λ > q k 1. Thus χ M s 1 /s1 (λ)+χ M {s 1,s 2 } /s 2 (λ)+ +χ M S /s n (λ) is positive for all λ > q k 1. It remains to consider χ M S(λ). Let S be the neck of uw, which is contained in M S. Clearly M S S = P G(r 1, q) for some r. Let E w be the bag corresponding to w. Let M 1 = M S (E w S ) and M 2 = M S \(E w S ). Then M 1 S = M 2 S. By [11, Corollary 6.9.6], S is a modular flat in M 1. By [11, Proposition ], M S is the generalized parallel connection of M 1 and M 2 across M 1 S. Since M has tree-width at most k, we know that Thus, by Theorem 2.2, r = r M S(S ) r M S(E w S ) = r M (E w ) k. χ M S(λ) = χ M 1 (λ)χ M2 (λ) χ P G(r 1,q)(λ). The denominator is positive for all λ > q r 1, hence it is positive for all λ > q k 1. By Lemma 1.3, since T has v(m) vertices, both M 1 and M 2 have rank less than r(m). By our induction hypothesis, both χ M1 (λ) and χ M2 (λ) are positive for all λ > q k 1. We may assume now that T has no leaf edge, hence T has a single vertex. Then r(m) k, since nw(e(m)) k, so let S be the set of elements in E(M q ) E(M). As before, χ M (λ) = χ M s 1 /s1 (λ) + χ M {s 1,s 2 } /s 2 (λ) + + χ M S /s n (λ) + χ M S(λ), and, as before, each term of this sum is positive for all λ > q k 1 with the possible exception of χ M S(λ). As M S is a projective geometry, χ M S(λ) = (λ 1)(λ q)(λ q 2 ) (λ q r 1 ). Thus χ M S(λ) > 0 for all λ > q r 1. As r k, we deduce that χ M (λ) > 0 for all λ > q k 1. This completes the proof. 10

11 4 Conclusion We noted earlier on that if we drop the restriction on representability then Theorem 1.4 is no longer true. Given that the line U 2,q is the simplest counterexample that we know of, it seems natural to ask the following question, as suggested by Geelen and Nelson [5]. Question 4.1. Let F be the class of loopless matroids having no minor isomorphic to U 2,q and tree-width at most k. Does c k,q exist such that the characteristic polynomial of any matroid in F has no real root larger than c k,q? References [1] Borgs, C. Absence of zeros for the chromatic polynomial on bounded degree graphs. Combinatorics, Probability and Computing 15, (2006). [2] Brylawski, T. Modular constructions for combinatorial geometries. Transactions of the American Mathematical Society 203, 1 44 (1975). [3] Brylawski, T., and Oxley, J. The Tutte polynomial and its applications. Matroid Applications, Encyclopedia of Mathematics and its Applications 40, Cambridge University Press, Cambridge (1992). [4] Dong, F., Koh, K., Teo, K. Chromatic polynomials and chromaticity of graphs, World Scientific Publishing Company, Singapore (2005). [5] Geelen, J., and Nelson, P. Private communication (2014). [6] Haggard, G., Pearce, D. J., and Royle, G. F. Computing Tutte polynomials. ACM Transactions on Mathematical Software 37, 24:1 24:17 (2010). [7] Hlilĕný, P. and Whittle, G. Matroid tree-width. European Journal of Combinatorics 27, (2006). [8] Jackson, B. A zero-free interval for chromatic polynomials of graphs. Combinatorics, Probability and Computing 2, (1993). 11

12 [9] Jacobsen, J. L., and Salas, J. Is the five-flow conjecture almost false? Journal of Combinatorial Theory Series B 103, (2013). [10] Mader, W. Homomorphieeigenschaften und mittlere Kantendichte von Graphen. Mathematische Annalen 174, (1967). [11] Oxley, J. Matroid theory, Second edition, Oxford University Press, New York (2011). [12] Robertson, N., and Seymour, P. D. Algorithmic aspects of tree-width. Journal of Algorithms 7, (1986). [13] Royle, G. F. Recent results on chromatic and flow roots of graphs and matroids. Surveys in Combinatorics, London Mathematical Society Lecture Note Series 365, Cambridge University Press, Cambridge (2009). [14] Sokal, A. D. Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Combinatorics, Probability and Computing 10, (2001). [15] Sokal, A. D., Chromatic roots are dense in the whole complex plane. Combinatorics, Probability and Computing, (2004). [16] Thomassen, C. The zero-free intervals for chromatic polynomials of graphs. Combinatorics, Probability and Computing 6, (1997). [17] Woodal, D. R. The largest real zero of the chromatic polynomial. Discrete Mathematics 172, (1997). [18] Zaslawsky, T., The Möbius function and the characteristic polynomial. Combinatorial Geometries, Encyclopedia of Mathematics and its Applications 29, Cambridge University Press, Cambridge (1990). 12