Removable edges in near‐bipartite bricks

• Published in: Journal of graph theory Vol. 108; no. 1; pp. 113 - 135
• Main Authors: Zhang, Yipei, Lu, Fuliang, Wang, Xiumei, Yuan, Jinjiang
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.01.2025
• Subjects: Apexes; brick; Bricks; Graph matching; Graph theory; Graphs; Lower bounds; near‐bipartite graph; perfect matching; removable edge
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.23173

An edge e of a matching covered graph G is removable if G − e is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. A nonbipartite matching covered graph G is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than K 4 and C 6 ¯ has at least Δ − 2 removable edges. A brick G is near‐bipartite if it has a pair of edges { e 1 , e 2 } such that G − { e 1 , e 2 } is a bipartite matching covered graph. In this paper, we show that in a near‐bipartite brick G with at least six vertices, every vertex of G, except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, G has at least | V ( G ) | − 6 2 removable edges. Moreover, all graphs attaining this lower bound are characterized.