On a Conjecture of Lovász Concerning Bricks: I. The Characteristic of a Matching Covered Graph

@article{Carvalho2002OnAC,
  title={On a Conjecture of Lov{\'a}sz Concerning Bricks: I. The Characteristic of a Matching Covered Graph},
  author={Marcelo H. de Carvalho and Claudio L. Lucchesi and Uppaluri S. R. Murty},
  journal={J. Comb. Theory, Ser. B},
  year={2002},
  volume={85},
  pages={94-136}
}
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… 
How to build a brick
A note on tight cuts in matching-covered graphs
TLDR
This note proves the following result which is slightly stronger than the conjecture: if a nontrivial tight cut of a matching covered graph $G$ is not an ELP-cut, then there is a sequence of matching covered graphs, such that for $i=1, 2,\ldots, r-1$, $G_i$ has an ELp-cut.
Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice
TLDR
It is proved that the number of double ears in an optimal ear decomposition of a matching covered graph G is b(G)+p(G), and that for any matching coveredgraph G, there is a basis for the matching lattice of G consisting of incidence vectors of perfect matchings of G.
Removable Edges in Near-bricks
TLDR
This paper generalizes the result obtained by Carvalho et al. (2002) on the existence of an edge whose deletion results in a graph with only one brick in its tight cut decomposition to the case of irreducible near-brick.
Birkhoff-von Neumann Graphs that are PM-Compact
TLDR
A complete characterization of matching covered graphs that are BvN as well as PMc is given, which is equivalent to the seemingly unrelated problem of deciding whether a given graph is $\overline{C_6}$-free.
On lower bounds on the number of perfect matchings in n-extendable bricks
TLDR
It is shown that every n-extendable brick with p vertices and q edges contains at least q − p + nn−1 perfect matchings.
M ay 2 01 7 On Two Unsolved Problems Concerning Matching Covered Graphs Dedicated to the memory
A cut C := ∂(X) of a matching covered graph G is a separating cut if both its C-contractions G/X and G/X are also matching covered. A brick is solid if it is free of nontrivial separating cuts. We
The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovasz and Plummer developed a
Matching Theory and Barnette's Conjecture
Barnette’s Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to
A Graph Theoretic Proof of the Tight Cut Lemma
TLDR
A new proof of the Tight Cut Lemma is presented, which attains both of the two reasonable features for the first time, namely, being {em purely graph theoretic} as well as {\em purely matching theory closed}.
...
...

References

SHOWING 1-6 OF 6 REFERENCES
Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice
TLDR
It is proved that the number of double ears in an optimal ear decomposition of a matching covered graph G is b(G)+p(G), and that for any matching coveredgraph G, there is a basis for the matching lattice of G consisting of incidence vectors of perfect matchings of G.
Ear Decompositions of Matching Covered Graphs
TLDR
It is shown 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.
Matching structure and the matching lattice
On a Conjecture of Lovász Concerning Bricks: II. Bricks of Finite Characteristic
TLDR
Here, it is established the existence of suitable separating cuts in nonsolid bricks and the theorem is proved by applying induction to cut-contractions with respect to such cuts.
The Two Ear Theorem on Matching-Covered Graphs
We give a simple and short proof for the two ear theorem on matching-covered graphs which is a well-known result of Lovasz and Plummer. The proof relies only on the classical results of Tutte and