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

A new lower bound on the number of perfect matchings in cubic graphs

International audienceWe prove that every n-vertex cubic bridgeless graph has at least n/2 perfect matchings and give a list of all 17 such graphs that have less than n/2+2 perfect matchings

How to build a brick

AbstractA graph is matching covered if it connected, has at least two vertices and each of its edges is contained in a perfect matching. A 3-connected graph G is a brick if, for any two vertices u and v of G, the graph G-{u,v} has a perfect matching. As shown by Lovász [Matching structure and the matching lattice, J. Combin. Theory Ser. B 43 (1987) 187–222] every matching covered graph G may be decomposed, in an essentially unique manner, into bricks and bipartite graphs known as braces. The number of bricks resulting from this decomposition is denoted by b(G).The object of this paper is to present a recursive procedure for generating bricks. We define four simple operations that can be used to construct new bricks from given bricks. We show that all bricks may be generated from three basic bricks K4, C¯6 and the Petersen graph by means of these four operations. In order to establish this, it turns out to be necessary to show that every brick G distinct from the three basic bricks has a thin edge, that is, an edge e such that (i) G-e is a matching covered graph with b(G-e)=1 and (ii) for each barrier B of G-e, the graph G-e-B has precisely |B|-1 isolated vertices, each of which has degree two in G-e. Improving upon a theorem proved in [M.H. de Carvalho, C.L. Lucchesi, U.S.R. Murty, On a conjecture of Lovász concerning bricks, I, The characteristic of a matching covered graph, J. Combin. Theory Ser. B 85 (2002) 94–136; M.H. de Carvalho, C.L. Lucchesi, U.S.R. Murty, On a conjecture of Lovász concerning bricks, II, Bricks of finite characteristic, J. Combin. Theory Ser. B 85 (2002) 137–180] we show here that every brick different from the three basic bricks has an edge that is thin.A cut of a matching covered graph G is separating if each of the two graphs obtained from G by shrinking the shores of the cut to single vertices is also matching covered. A brick is solid if it does not have any nontrivial separating cuts. Solid bricks have many interesting properties, but the complexity status of deciding whether a given brick is solid is not known. Here, by using our theorem on the existence of thin edges, we show that every simple planar solid brick is an odd wheel

Brick Generation and Conformal Subgraphs

A nontrivial connected graph is matching covered if each of its edges lies in a perfect matching. Two types of decompositions of matching covered graphs, namely ear decompositions and tight cut decompositions, have played key roles in the theory of these graphs. Any tight cut decomposition of a matching covered graph results in an essentially unique list of special matching covered graphs, called bricks (which are nonbipartite and 3-connected) and braces (which are bipartite). A fundamental theorem of LovU+00E1sz (1983) states that every nonbipartite matching covered graph admits an ear decomposition starting with a bi-subdivision of $K_4$ or of the triangular prism $\overline{C_6}$. This led Carvalho, Lucchesi and Murty (2003) to pose two problems: (i) characterize those nonbipartite matching covered graphs which admit an ear decomposition starting with a bi-subdivision of $K_4$, and likewise, (ii) characterize those which admit an ear decomposition starting with a bi-subdivision of $\overline{C_6}$. In the first part of this thesis, we solve these problems for the special case of planar graphs. In Chapter 2, we reduce these problems to the case of bricks, and in Chapter 3, we solve both problems when the graph under consideration is a planar brick. A nonbipartite matching covered graph G is near-bipartite if it has a pair of edges U+03B1 and U+03B2 such that G-{U+03B1,U+03B2} is bipartite and matching covered; examples are $K_4$ and $\overline{C_6}$. The first nonbipartite graph in any ear decomposition of a nonbipartite graph is a bi-subdivision of a near-bipartite graph. For this reason, near-bipartite graphs play a central role in the theory of matching covered graphs. In the second part of this thesis, we establish generation theorems which are specific to near-bipartite bricks. Deleting an edge e from a brick G results in a graph with zero, one or two vertices of degree two, as G is 3-connected. The bicontraction of a vertex of degree two consists of contracting the two edges incident with it; and the retract of G-e is the graph J obtained from it by bicontracting all its vertices of degree two. The edge e is thin if J is also a brick. Carvalho, Lucchesi and Murty (2006) showed that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a thin edge. In general, given a near-bipartite brick G and a thin edge e, the retract J of G-e need not be near-bipartite. In Chapter 5, we show that every near-bipartite brick G, distinct from $K_4$ and $\overline{C_6}$, has a thin edge e such that the retract J of G-e is also near-bipartite. Our theorem is a refinement of the result of Carvalho, Lucchesi and Murty which is appropriate for the restricted class of near-bipartite bricks. For a simple brick G and a thin edge e, the retract of G-e may not be simple. It was established by Norine and Thomas (2007) that each simple brick, which is not in any of five well-defined infinite families of graphs, and is not isomorphic to the Petersen graph, has a thin edge such that the retract J of G-e is also simple. In Chapter 6, using our result from Chapter 5, we show that every simple near-bipartite brick G has a thin edge e such that the retract J of G-e is also simple and near-bipartite, unless G belongs to any of eight infinite families of graphs. This is a refinement of the theorem of Norine and Thomas which is appropriate for the restricted class of near-bipartite bricks