Faces of Birkhoff Polytopes

The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics. In this paper we study combinatorial types L of faces of a Birkhoff polytope. The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face with combinatorial type L. By a result of Billera and Sarangarajan, a combinatorial type L of a d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is at most d. We will characterize those types whose Birkhoff dimension is at least 2d-3, and we prove that any type whose Birkhoff dimension is at least d is either a product or a wedge over some lower dimensional face. Further, we computationally classify all d-dimensional combinatorial types for d between 2 and 8.Comment: 29 page

Constructive Characterization of Critical Bipartite Grafts

Factor-critical graphs are a classical concept in matching theory that constitute an important component of the Gallai-Edmonds canonical decomposition and Edmonds' algorithm for maximum matchings. Lov\'asz provided a constructive characterization of factor-critical graphs in terms of ear decompositions. This characterization has been a useful inductive tool for studying factor-critical graphs and also connects them with Edmonds' algorithm. Joins in grafts, also known as $T$-joins in graphs, are a classical variant of matchings proposed in terms of parity. Minimum joins and grafts are generalizations of perfect matchings and graphs with perfect matchings, respectively. Accordingly, graft analogues of fundamental concepts and results from matching theory, such as canonical decompositions, will develop the theory of minimum join. In this paper, we propose a new concept, critical quasicombs, as a bipartite graft analogue of factor-critical graphs and provide a constructive characterization of critical quasicombs using a graft version of ear decompositions. This characterization can be considered as a bipartite graft analogue of Lov\'asz' result. From our results, the Dulmage-Mendelsohn canonical decomposition, originally a theory for bipartite graphs, has been generalized for bipartite grafts.Comment: Part of results from arXiv:2101.06678 in stand-alone and revised for

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest number $s$ with this property the index of depth stability of $I$ and denotes it by $\operatorname{dstab}(I)$. This invariant remains mysterious til now. The main result of this paper gives an explicit formula for $\operatorname{dstab}(I)$ when $I$ is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize $\operatorname{dstab}(I)$ explicitly. The formula expresses $\operatorname{dstab}(I)$ in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.Comment: 23 pages, 4 figure