Tutte polynomials for counting and classifying orbits

Given a graph Γ and an automorphism group , we define some polynomials which count and classify the orbits of G on various structures on Γ, as counted by the Tutte polynomial, while also specialising to the Tutte polynomial.     

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105-119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57-65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique.     

A Characterization of the Tutte Polynomial via Combinatorial Embeddings

We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to activities. Tutte’s notion of activity requires a choice of a linear order on the edge set (though the generating function of the activities is, in fact, independent of this order). We define a new notion of activity, the embedding-activity, which requires a choice of a combinatorial embedding of the graph, that is, a cyclic order of the edges around each vertex. We prove that the Tutte polynomial equals the generating function of spanning trees counted according to embedding-activities. This generating function is, in fact, independent of the embedding. Received March 15, 2006     

Triangulations of Cayley and Tutte polytopes

Cayley polytopes were defined recently as convex hulls of Cayley compositions introduced by Cayley in 1857. In this paper we resolve Braun’s conjecture  , which expresses the volume of Cayley polytopes in terms of the number of connected graphs. We extend this result to two one-variable deformations of Cayley polytopes (which we call t$t$-Cayley   and t$t$-Gayley polytopes), and to the most general two-variable deformations, which we call Tutte polytopes. The volume of the latter is given via an evaluation of the Tutte polynomial of the complete graph.     

On Graphs Determined by Their Tutte Polynomials

We say that a graph G is T-unique if any other graph having the same Tutte polynomial as G is necessarily isomorphic to G. In this paper we show that several well-known families of graphs are T-unique: wheels, squares of cycles, complete multipartite graphs, ladders, Möbius ladders, and hypercubes. In order to prove these results, we show that several parameters of a graph, like the number of cycles of length 3, 4 and 5, and the edge-connectivity are determined by its Tutte polynomial.Research partially supported by projects BFM2001-2340 and by CUR Gen. Cat. 1999SGR00356Final version received: January 10, 2003     

Is the critical percolation probability local?

We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. We also prove a finite analogue of this statement, valid for expander graphs, without any girth assumption.     

Contraction–Deletion Invariants for Graphs

We consider generalizations of the Tutte polynomial on multigraphs obtained by keeping the main recurrence relation T(G)=T(G/e)+T(G−e) for eE(G) neither a bridge nor a loop and dropping the relations for bridges and loops. Our first aim is to find the universal invariant satisfying these conditions, from which all others may be obtained. Surprisingly, this turns out to be the universal V-function Z of Tutte (1947, Proc. Cambridge Philos. Soc.43, 26–40) defined to obey the same relation for bridges as well. We also obtain a corresponding result for graphs with colours on the edges and describe the universal coloured V-function, which is more complicated than Z. Extending results of Tutte (1974, J. Combin. Theory Ser. B16, 168–174) and Brylawski (1981, J. Combin. Theory Ser. B30, 233–246), we give a simple proof that there are non-isomorphic graphs of arbitrarily high connectivity with the same Tutte polynomial and the same value of Z. We conjecture that almost all graphs are determined by their chromatic or Tutte polynomials and provide mild evidence to support this.     

Counting colored planar maps: Algebraicity results

We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial χM(q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q≠0,4 is of the form 2+2cos(jπ/m), for integers j and m. This includes the two integer values q=2 and q=3. We extend this to planar maps weighted by their Potts polynomial PM(q,ν), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two “catalytic” variables. To our knowledge, this is the first time such equations are being solved since Tutte?s remarkable solution of properly q-colored triangulations.     

On chromatic and flow polynomial unique graphs

It is known that the chromatic polynomial and flow polynomial of a graph are two important evaluations of its Tutte polynomial, both of which contain much information of the graph. Much research is done on graphs determined entirely by their chromatic polynomials and Tutte polynomials, respectively. Oxley asked which classes of graphs or matroids are determined by their chromatic and flow polynomials together. In this paper, we found several classes of graphs with this property. We first study which graphic parameters are determined by the flow polynomials. Then we study flow-unique graphs. Finally, we show that several classes of graphs, ladders, Möbius ladders and squares of n-cycle are determined by their chromatic polynomials and flow polynomials together. A direct consequence of our theorem is a result of de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomial, Graphs Comb. 20 (2004) 105-119] that these classes of graphs are Tutte polynomial unique.     

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.     

On the reconstruction of graph invariants

The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U-polynomial, the universal edge elimination polynomial ξ and the colored versions of the latter two are reconstructible.We also present a method of reconstructing boolean graph invariants, or in other words, proving recognizability of graph properties (of colored or uncolored graphs), using first order logic.     

Invariants of Composite Networks Arising as a Tensor Product

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves.      

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.     

A note on Tutte polynomials and Orlik–Solomon algebras

Let be a (central) arrangement of hyperplanes in and the dependence matroid of the linear forms . The Orlik–Solomon algebra of a matroid is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The graded algebra is isomorphic to the cohomology algebra of the manifold . The Tutte polynomial is a powerful invariant of the matroid . When is a rank 3 matroid and the θ_Hi are complexifications of real linear forms, we will prove that determines . This result partially solves a conjecture of Falk.     

Preface: Old and New Perspectives on the Tutte Polynomial

This paper introduces a special issue on the Tutte polynomial derived from the Second Workshop on Tutte Polynomials and Applications, 2005, held at the Centre de Recerca Matemàtica, Bellaterra, Catalonia. We discuss the prehistory of Tutte polynomials and two current areas of research, to what extent a graph is determined by its chromatic or Tutte polynomial and generic versions of Tutte polynomials. Received February 28, 2007     

The Tutte–Potts connection in the presence of an external magnetic field

The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V-polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The V-polynomial generalizes Noble and Welshʼs W-polynomial, which extends the Tutte polynomial by incorporating vertex weights and adapting contraction to accommodate them. We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the V-polynomial, and hence a polynomial with deletion–contraction reduction and Fortuin–Kasteleyn type representation. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including complexity results, to bear on a wider range of statistical mechanics models.     

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or morphism, or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into Boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into Boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set, and applies to graph and directed graph homomorphisms on a linearly ordered edge-set.     

Extensions of barrier sets to nonzero roots of the matching polynomial

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to their connection to maximum matchings in a graph. For a root θ of the matching polynomial, we define θ-barrier and θ-extreme sets. We prove a generalized Berge-Tutte formula and give a characterization for the set of all θ-special vertices in a graph.     

Small subgraphs of random regular graphs

The main aim of this short paper is to answer the following question. Given a fixed graph H, for which values of the degree d does a random d-regular graph on n vertices contain a copy of H with probability close to one?     

Edge-Colorings of 4-Regular Graphs with the Minimum Number of Palettes

A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Horňák, Kalinowski, Meszka and Wo?niak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs. In particular, we show that certain values of the palette index imply the existence of an even cycle decomposition of size 3 (a partition of the edge-set of a graph into 3 2-regular subgraphs whose connected components are cycles of even length). This result can be extended to 4d-regular graphs. Moreover, in studying the palette index of a 4-regular graph, the following problem arises: does there exist a 4-regular graph whose even cycle decompositions cannot have size smaller than 4?