{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T18:38:15Z","timestamp":1781635095499,"version":"3.54.5"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Lov\u00e1sz (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.)\r\nA 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\u00e1sz which states that every brick, distinct from $K_4$, $\\overline{C_6}$ and the Petersen graph, has a $b$-invariant edge.\r\nA 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.\r\nWe 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.<\/jats:p>","DOI":"10.37236\/8594","type":"journal-article","created":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T09:18:32Z","timestamp":1579857512000},"source":"Crossref","is-referenced-by-count":8,"title":["On Essentially 4-Edge-Connected Cubic Bricks"],"prefix":"10.37236","volume":"27","author":[{"given":"Nishad","family":"Kothari","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marcelo H.","family":"De Carvalho","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cl\u00e1udio L.","family":"Lucchesi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Charles H. C.","family":"Little","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2020,1,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p22\/8007","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p22\/8007","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,24]],"date-time":"2020-01-24T09:18:32Z","timestamp":1579857512000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,24]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8594","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1,24]]},"article-number":"P1.22"}}