Marcelo H. de Carvalho

Ear Decompositions of Matching Covered Graphs

G different from and has at least Δ edge-disjoint removable ears, where Δ is the maximum degree of G. This shows that any matching covered graph G has at least Δ! different ear decompositions, and thus is a generalization of the fundamental theorem of Lovász and Plummer establishing the existence of ear decompositions. We also show that every brick G different from and has Δ− 2 edges, each of which is a removable edge in G, that is, an edge whose deletion from G results in a matching covered graph. This generalizes a well-known theorem of Lovász. We also give a simple proof of another theorem due to Lovász which says that every nonbipartite matching covered graph has a canonical ear decomposition, that is, one in which either the third graph in the sequence is an odd-subdivision of or the fourth graph in the sequence is an odd-subdivision of . Our method in fact shows that every nonbipartite matching covered graph has a canonical ear decomposition which is optimal, that is one which has as few double ears as possible. Most of these results appear in the Ph. D. thesis of the first author [1], written under the supervision of the second author.

On a Conjecture of Lovász Concerning Bricks

In 1987, Lovasz conjectured that every brick G different from K4, C6, and the Petersen graph has an edge e such that G?e is a matching covered graph with exactly one brick. Lovasz and Vempala announced a proof of this conjecture in 1994. Their paper is under preparation. In this paper and its sequel (2001, J. Combin. Theory. Ser. B) we present a proof of this conjecture. We shall in fact prove that if G is a brick different from K4, C6, R8 that does not have the Petersen graph as its underlying simple graph, then it has two edges e and f such that both G?e and G?f are matching covered graphs with exactly one brick, with the additional property that, in each case, the underlying simple graph of that one brick is different from the Petersen graph. A cut C of a matching covered graph G is a separating cut if the two C -contractions of G are matching covered. In this paper, we introduce the notion of the characteristic of a separating cut in a matching covered graph and establish some basic properties. We use those properties to first prove our theorem for solid bricks, that is, bricks which do not have any nontrivial separating cuts. The proof of the theorem for nonsolid bricks will be presented in the sequel.

On a Conjecture of Lovász Concerning Bricks: II. Bricks of Finite Characteristic

Abstract In (2001, J. Combin. Theory Ser. B ) we established the validity of the main theorem (1.1) for solid bricks. Here, we establish the existence of suitable separating cuts in nonsolid bricks and prove the theorem by applying induction to cut-contractions with respect to such cuts.

The perfect matching polytope and solid bricks

The perfect matching polytope of a graph G is the convex hull of the set of incidence vectors of perfect matchings of G. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69B 1965 125) showed that a vector x in QE belongs to the perfect matching polytope of G if and only if it satisfies the inequalities: (i) x ≥ 0 (non-negativity), (ii) x(∂(v)) = 1, for all v ∈ V (degree constraints) and Off) x(∂(S)) ≥ 1, for all odd subsets S of V (odd set constraints). In this paper, we characterize graphs whose perfect matching polytopes are determined by non-negativity and the degree constraints. We also present a proof of a recent theorem of Reed and Wakabayashi.

A CHARACTERIZATION OF PM-COMPACT CLAW-FREE CUBIC GRAPHS

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. This paper characterizes claw-free cubic graphs whose 1-skeleton graphs of perfect matching polytopes have diameter 1.

A characterization of PM-compact bipartite and near-bipartite graphs

Abstract The perfect-matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G . This paper characterizes bipartite and nearly bipartite graphs whose 1-skeleton graphs of perfect-matching polytopes have diameter 1.

Matching Covered Graphs and Subdivisions ofK4andformula

We give a very simple proof that every non-bipartite matching covered graph contains a nice subgraph that is an odd subdivision ofK4orformula]. It follows immediately that every brick different fromK4andformula]has an edge whose removal preserves the matching covered property. These are classical and very useful results due to Lovasz.

On the Number of Perfect Matchings in a Bipartite Graph

In this paper we show that, with 11 exceptions, any matching covered bipartite graph on

How to build a brick

n

An O ( VE ) algorithm for ear decompositions of matching-covered graphs

vertices, with minimum degree greater than two, has at least