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Some Inequalities for the Tutte Polynomial

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Some Inequalities for the Tutte Polynomial View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 32 (2011) 422–433 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Some inequalities for the Tutte polynomial Laura E. Chávez-Lomelí a, Criel Merino b,1, Steven D. Noble c, Marcelino Ramírez-Ibáñez b a Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Avenida San Pablo 180, colonia Reynosa Tamaulipas, Delegación Azcapotzalco, Mexico b Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de la Investigación Científica, Circuito Exterior, C.U. Coyoacán 04510, México D.F., Mexico c Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge UB8 3PH, UK article info a b s t r a c t Article history: We prove that the Tutte polynomial of a coloopless paving matroid Received 22 February 2010 is convex along the portion of the line x C y D p lying in the Accepted 6 September 2010 positive quadrant. Every coloopless paving matroid is in the class Available online 29 December 2010 of matroids which contain two disjoint bases or whose ground set is the union of two bases. For this latter class we give a proof that TM .a; a/ ≤ maxfTM .2a; 0/; TM .0; 2a/g for a ≥ 2. We conjecture that TM .1; 1/ ≤ maxfTM .2; 0/; TM .0; 2/g for the same class of matroids. We also prove this conjecture for some families of graphs and matroids. ' 2010 Elsevier Ltd. All rights reserved. 1. Introduction The Tutte polynomial is a two-variable polynomial which can be defined for a graph G or more generally a matroid M. The Tutte polynomial has many interesting combinatorial interpretations when evaluated at different points (x; y) and along several algebraic curves. For example, for a graph G, the Tutte polynomial along the line y D 0 is the chromatic polynomial, after a suitable change of variable and multiplication by an easy term. In similar ways, we can get the flow polynomial of a graph, the all terminal reliability of a network and the partition function of the Q -state Potts model. When considering a GF(q)-representable matroid, the Tutte polynomial gives us the weight enumerator of linear codes over GF(q) associated to M. All the necessary background on the Tutte polynomial is contained in Section 2. E-mail addresses: [email protected] (L.E. Chávez-Lomelí), [email protected], [email protected] (C. Merino), [email protected] (S.D. Noble), [email protected] (M. Ramírez-Ibáñez). 1 IMATE OAXACA, León 2, altos, colonia centro, C.P. 68000, Oaxaca de Juárez, Mexico. 0195-6698/$ – see front matter ' 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2010.11.005 L.E. Chávez-Lomelí et al. / European Journal of Combinatorics 32 (2011) 422–433 423 For a convex set S, a function f V S ! R is a convex function if for all x1; x2 2 S and t 2 .0; 1/; f .tx1 C .1 − t/x2/ ≤ tf .x1/ C .1 − t/f .x2/. In this work, we mainly concentrate on proving the convexity of one-variable polynomials but in the conclusion we also consider the convexity of two- variable polynomials over the convex set positive quadrant. It is well known [3] that the Tutte polynomial of a matroid M has an expansion X i j TM .x; y/ D tijx y ; i;j in which each coefficient tij is non-negative. Consequently, for m ≥ 0 and for any b; TM .x; y/ increases along the portion of the line y D mxCb lying in the positive quadrant, as x increases. The simplicity in the behaviour of T along lines with positive gradient suggests the study of the behaviour of TM along lines with negative gradient in the positive quadrant. Merino and Welsh [15] were the first to consider this and were particularly interested in resolving the question of whether the Tutte polynomial is convex along the portion of the line x C y D 2 lying in the positive quadrant. They made the following intriguing conjecture. Conjecture 1.1. Let G be a 2-connected graph with no loops. Then maxfTM.G/.2; 0/; TM.G/.0; 2/g ≥ TM.G/.1; 1/: (1) Notice that this is a necessary condition for T to be a convex function along the portion of the line mentioned above. Any graph with at least one loop and at least one coloop fails to satisfy (1), so (1) cannot hold for all graphs. The main reason for the particular interest in the points .2; 0/; .0; 2/ and .1; 1/ is that in a connected graph G; TM.G/.2; 0/; TM.G/.0; 2/ and TM.G/.1; 1/ give the number of acyclic orientations, totally cyclic orientations and spanning trees in G, respectively. Definitions of acyclic and totally cyclic orientations are contained in Section 2. A related question is to determine whether any loopless and bridgeless graph G satisfies the apparently stronger requirement 2 TM.G/.2; 0/TM.G/.0; 2/ ≥ .TM.G/.1; 1// : Relatively little progress has been made to resolve these questions. However, Jackson in [11] has shown, with a clever argument, that for any loopless and bridgeless graph G, and a and b real positive numbers with b ≥ a.a C 2/, TM.G/.b; 0/TM.G/.0; b/ ≥ TM.G/.a; a/: In this paper, we make three contributions. First, in Section 4, we show in Theorem 4.9 that the Tutte polynomial T of a coloopless paving matroid satisfies the inequality tT .x1; y1/ C .1 − t/T .x2; y2/ ≥ T .tx1 C .1 − t/x2; ty1 C .1 − t/y2/; (2) where 0 ≤ t ≤ 1 and x1; x2; y1; y2 are non-negative and satisfy x1 C y1 D x2 C y2. That is, T is convex along the portion of the line x C y D p lying in the positive quadrant. A paving matroid is one in which all circuits have size at least r.M/. Interest in them stems from a conjecture in [13] which says that asymptotically every matroid is paving. The special case of (2), obtained by setting x1 D y2 D 2; x2 D y1 D 0 and t D 1=2, establishes (1) for the class of paving matroids. Therefore if the above conjecture is true then we have established (1) for, asymptotically all coloopless matroids. Second, in Section 5, we prove that (1) holds for some smaller classes of matroids and graphs that are not paving matroids. Finally, in Section 3, we prove that if the ground set of M contains two disjoint bases then TM .0; 2a/ ≥ TM .a; a/, whenever a ≥ 2, and dually if the ground set of M is the union of two bases then TM .2a; 0/ ≥ TM .a; a/. These results cannot be obtained with the methods used by Jackson in [11]. We conclude with a brief discussion of the natural question, for which matroids is TM a convex function in the positive quadrant? 424 L.E. Chávez-Lomelí et al. / European Journal of Combinatorics 32 (2011) 422–433 2. Preliminaries We assume that the reader has some familiarity with matroid and graph theory. For matroid theory we follow Oxley's book [18] and for graph theory we follow Diestel's book [8]. The Tutte polynomial is a matroid invariant over the ring ZTx; yU. Further details of many of the concepts treated here can be found in [23,5]. Some of the richness of the Tutte polynomial is due to its numerous equivalent definitions. One of the simplest definitions, which is often the easiest way to prove properties of the Tutte polynomial, uses the notion of rank. If M D .E; r/ is a matroid, where r is the rank-function of M, and A ⊆ E, we denote r.E/ − r.A/ by z.A/ and jAj − r.A/ by n.A/. Definition 2.1. The Tutte polynomial of M; TM .x; y/, is defined as follows: X z.A/ n.A/ TM .x; y/ D .x − 1/ .y − 1/ : (3) A⊆E Almost immediately, we see that TM .1; 1/ equals the number of bases of M and TM .2; 2/ equals 2jEj. Recall that if M D .E; r/ is a matroid, then M∗ D .E; r∗/ is its dual matroid, where r∗.A/ D jAj − r.E/ C r.E n A/. Because zM∗ .A/ D nM .E n A/ and nM∗ .A/ D zM .E n A/ it follows that TM .x; y/ D TM∗ .y; x/. For a graphic matroid M.G/, the evaluations of the Tutte polynomial at .2; 0/ and .0; 2/ equal the number of acyclic orientations and the number of totally cyclic orientations of G, respectively. An acyclic orientation of a graph G is an orientation where there are no directed cycles. A totally cyclic orientation is an orientation where every edge is in a directed cycle. See [5] for a proof of this result. ∗ We let α.G/ and α .G/ denote TM.G/.2; 0/ and TM.G/.0; 2/, respectively. If G is connected, the number of spanning trees of G is the evaluation of the Tutte polynomial at .1; 1/ and this quantity is denoted by τ.G/. The Tutte polynomial may be also defined by a linear recursion relation given by deleting and contracting elements that are neither loops nor coloops. Definition 2.2.
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