bb-invariant edges in essentially 4-edge-connected near-bipartite cubic bricks

A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is {\em essentially 4-edge-connected} if it does not contain nontrivial 3-cuts. A brick $G$ is {\em near-bipartite} if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered. Kothari, de Carvalho, Lucchesi and Little proved 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. Moreover, they made a conjecture: every essentially 4-edge-connected cubic near-bipartite brick $G$, distinct from $K_4$, has at least $|V(G)|/2$ $b$-invariant edges. We confirm the conjecture in this paper. Furthermore, all the essentially 4-edge-connected cubic near-bipartite bricks, the numbers of $b$-invariant edges of which attain the lower bound, are presented

On essentially 4-edge-connected cubic bricks

Lov\'asz (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 Lov\'asz 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.Comment: Accepted for publication in Electronic Journal of Combinatorics (December 2019