Abstract
Lovász (1987) proved that every matching covered graph $G$ may be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let $b(G)$ denote the number of bricks. An edge $e$ is removable if $G-e$ is also matching covered; furthermore, $e$ is $b$-invariant if $b(G-e)=1$, and $e$ is quasi-$b$-invariant if $b(G-e)=2$. (Each edge of the Petersen graph is quasi-$b$-invariant.)
 A brick $G$ is near-bipartite if it has a pair of edges $\{e,f\}$ so that $G-e-f$ is matching covered and bipartite; such a pair $\{e,f\}$ is a removable doubleton. (Each of $K_4$ and the triangular prism $\overline{C_6}$ has three removable doubletons.) Carvalho, Lucchesi and Murty (2002) proved a conjecture of Lovász which states that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a $b$-invariant edge.
 A cubic graph is essentially $4$-edge-connected if it is $2$-edge-connected and if its only $3$-cuts are the trivial ones; it is well-known that each such graph is either a brick or a brace; we provide a graph-theoretical proof of this fact.
 We prove that if $G$ is any essentially $4$-edge-connected cubic brick then its edge-set may be partitioned into three (possibly empty) sets: (i) edges that participate in a removable doubleton, (ii) $b$-invariant edges, and (iii) quasi-$b$-invariant edges; our Main Theorem states that if $G$ has two adjacent quasi-$b$-invariant edges, say $e_1$ and $e_2$, then either $G$ is the Petersen graph or the (near-bipartite) Cubeplex graph, or otherwise, each edge of $G$ (distinct from $e_1$ and $e_2$) is $b$-invariant. As a corollary, we deduce that each essentially $4$-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges.
Highlights
We prove that if G is any essentially 4-edge-connected cubic brick its edgeset may be partitioned into three sets: (i) edges that participate in a removable doubleton, (ii) b-invariant edges, and (iii) quasi-b-invariant edges; our Main Theorem states that if G has two adjacent quasi-b-invariant edges, say e1 and e2, either G is the Petersen graph or the Cubeplex graph, or otherwise, each edge of G is b-invariant
Tutte [26] proved his celebrated 1-factor Theorem characterizing matchable graphs, and deduced as a corollary that in a 2-edge-connected cubic graph each edge lies in a perfect matching
The triangular prism C6 has a nontrivial 3-cut C, and for each e ∈ C, the edge e is neither removable nor does it participate in a removable doubleton
Summary
Tutte [26] proved his celebrated 1-factor Theorem characterizing matchable graphs, and deduced as a corollary that in a 2-edge-connected cubic graph each edge lies in a perfect matching. For distinct vertices u and v of G, it is deduced from Tutte’s Theorem that the graph G − u − v is matchable if and only if no barrier of G contains both u and v. An edge e of G is admissible if there is some perfect matching of G that contains e; otherwise it is inadmissible. An edge e is admissible if and only if no barrier of G contains both ends of e. A connected graph with two or more vertices is matching covered if each of its edges is admissible. A cubic graph is matching covered if and only if it is 2-edge-connected
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