2024/06/01 by Zhang, Yipei, Lu, Fuliang, Wang, Xiumei +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2406.00292
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 K4 and C6 has at least Δ-2 removable edges. A brick G is near-bipartite if it has a pair of edges \e1,e2\ such that G-\e1,e2\ 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.