# 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…

## 41 Citations

A note on tight cuts in matching-covered graphs

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2021

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

- MathematicsJ. Comb. Theory, Ser. B
- 2002

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

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2013

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

- MathematicsSIAM J. Discret. Math.
- 2020

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

- MathematicsAustralas. J Comb.
- 2002

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

- Mathematics
- 2017

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

- Mathematics
- 2018

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

- MathematicsArXiv
- 2022

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

- Mathematics, Computer Science
- 2015

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

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Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice

- MathematicsJ. Comb. Theory, Ser. B
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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

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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.

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

- MathematicsJ. Comb. Theory, Ser. B
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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.

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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…