{"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":1781635095738,"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>A brick\u00a0is a\u00a0 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\u00a0$b$-invariant\u00a0if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is\u00a0essentially 4-edge-connected\u00a0if it does not contain nontrivial 3-cuts. A brick $G$ is\u00a0near-bipartite\u00a0if it has a pair of edges $\\{e_1, e_2\\}$ such that $G-\\{e_1,e_2\\}$ is bipartite and matching covered.\r\nKothari, de Carvalho, Lucchesi\u00a0 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.<\/jats:p>","DOI":"10.37236\/8945","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:12Z","timestamp":1584578652000},"source":"Crossref","is-referenced-by-count":3,"title":["$b$-Invariant Edges in Essentially 4-Edge-Connected Near-Bipartite Cubic Bricks"],"prefix":"10.37236","volume":"27","author":[{"given":"Fuliang","family":"Lu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xing","family":"Feng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2020,3,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p55\/8043","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i1p55\/8043","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:44:13Z","timestamp":1584578653000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i1p55"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/8945","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,20]]},"article-number":"P1.55"}}