Combinatorial evaluations of the Tutte polynomial

The Tutte polynomial is one of the most important and most useful invariants of a graph. It was discovered as a two variable generalization of the chromatic polynomial [15, 16], and has been studied in literally hundreds of papers, in part due to its connections to various fields ranging from Enumerative Combinatorics to Knot Theory, from Statistical Physics to Computer Science. We refer the reader to [3] for a nice introduction to the subject and to [18] for a well written and extensive survey of modern theory and applications. The main subject of this paper is combinatorial evaluations of the Tutte polynomial, by which we mean combinatorial interpretations of its values. The reader may recall classical evaluations in terms of the number of proper (vertex) colorings, spanning trees, spanning subgraphs, acyclic orientations, etc. We show that when the graph is planar, certain other values of the Tutte polynomial have a number of combinatorial interpretations specific to the plane embedding. We give new combinatorial evaluations in terms of two different edge colorings, claw coverings, and, for particular graphs on a square grid, in terms of Wang tilings and T-tetromino tilings. It is natural to ask which of our constructions extend to nonplanar graphs, in a situation when the graph is embedded into a surface of higher genus. We discover that natural generalizations of our results give combinatorial evaluations not of the Tutte polynomial, but of a specialization of a recently introduced Bollobás-Riordan polynomial [4, 5]. We should mention here that there are certain limitations of this approach; many, but not all of our results extend to that case. This paper is a sequel to our previous work [10], but can be read independently as it does not use any of our previous results. In fact, the paper is completely self-contained. Basically, in [10] we found an unexpected connection between the number of T-tetromino tilings of rectangular shape regions and the evaluation of the Tutte polynomial at (3, 3). Here we undertake a thorough investigation of this phenomenon, and explain it by a series of bijections and combinatorial evaluations. One can view this paper as a generalization of the results in [10] presented in a different language and with a philosophy opposite to