Bernardi gave a formula for the Tutte polynomial T (x, y) of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial I is a generalization of T (x, 1) to hypergraphs. We supply a Bernardi-type description of I using a ribbon structure on the underlying bipartite graph G. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of G in the same way as I is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses), constructed by Bernardi and further studied by Baker and Wang, between spanning trees and break divisors.

A famous conjecture (usually called Ryser's conjecture) that appeared in the PhD thesis of his student, J. R. Henderson [15], states that for an r-uniform r-partite hypergraph H, the inequality τ (H) ≤ (r − 1)·ν(H) always holds.This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig's theorem [18], and in the case of r = 3, which was proved by Aharoni in 2001 [3].Here we study some special cases of Ryser's conjecture. First of all, the most studied special case is when H is intersecting. Even for this special case, not too much is known: this conjecture is proved only for r ≤ 5 in [10,21]. For r > 5 it is also widely open.Generalizing the conjecture for intersecting hypergraphs, we conjecture the following. If an r-uniform r-partite hypergraph H is t-intersecting (i.e., every two hyperedges meet in at least t < r vertices), then τ (H) ≤ r − t. We prove this conjecture for the case t > r/4. Gyárfás [10] showed that Ryser's conjecture for intersecting hypergraphs is equivalent to saying that the vertices of an r-edge-colored complete graph can be covered by r − 1 monochromatic components.Motivated by this formulation, we examine what fraction of the vertices can be covered by r − 1 monochromatic components of different colors in an r-edgecolored complete graph. We prove a sharp bound for this problem.Finally we prove Ryser's conjecture for the very special case when the maximum degree of the hypergraph is two.

Baker and Norine introduced a graph-theoretic analogue of the Riemann-Roch theory. A central notion in this theory is the rank of a divisor. In this paper we prove that computing the rank of a divisor on a graph is NP-hard, even for simple graphs.The determination of the rank of a divisor can be translated to a question about a chip-firing game on the same underlying graph. We prove the NP-hardness of this question by relating chip-firing on directed and undirected graphs.

We prove several combinatorial properties of suffix arrays, including a characterization of suffix arrays through a bijection with a certain well-defined class of permutations. Our approach is based on the characterization of Burrows-Wheeler arrays given in [1], that we apply by reducing suffix sorting to cyclic shift sorting through the use of an additional sentinel symbol. We show that the characterization of suffix arrays for a special case of binary alphabet given in [2] easily follows from our characterization. Based on our results, we also provide simple proofs for the enumeration results for suffix arrays, obtained in [3]. Our approach to characterizing suffix arrays is the first that exploits their relationship with Burrows-Wheeler permutations.

Recently O. Bernardi gave a formula for the Tutte polynomial T (x, y) of a graph, based on spanning trees and activities just like the original definition, but using a fixed ribbon structure to order the set of edges in a different way for each tree. The interior polynomial I is a generalization of T (x, 1) to hypergraphs. We supply a Bernardi-type description of I using a ribbon structure on the underlying bipartite graph G. Our formula works because it is determined by the Ehrhart polynomial of the root polytope of G in the same way as I is. To prove this we interpret the Bernardi process as a way of dissecting the root polytope into simplices, along with a shelling order. We also show that our generalized Bernardi process gives a common extension of bijections (and their inverses) constructed by Baker and Wang between spanning trees and break divisors.