The Number of Vertices of Degree k in a Minimally k-Edge-Connected Graph

Let G be a minimally k-edge-connected simple graph and u(G) be the number of vertices of degree k in G. It is proved that (i) u(G) ≥ ((2k - 1)/2(2k + 3)) |G| + (14k + l)/2(2k+3) for even k ≥ 6 and u(G) ≥ |G|/4 + 13/4 for k = 4, and (ii) u(G) ≥ ((2k − 1)/2(2k + 5)) |G| + (10k + l)/(2k + 5) for odd k ≥ 7 and u(G) ≥ |G|/5 + 24/5 for k = 5, where |G| denotes the number of vertices of G.