# 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}
}
• Published 1 May 2002
• Mathematics
• J. Comb. Theory, Ser. B
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…
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## References

SHOWING 1-6 OF 6 REFERENCES
Optimal Ear Decompositions of Matching Covered Graphs and Bases for the Matching Lattice
• Mathematics
J. 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.
Ear Decompositions of Matching Covered Graphs
• Mathematics
Comb.
• 1999
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
• Mathematics
J. Comb. Theory, Ser. B
• 2002
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