Sufficient conditions for -optimality in triangle-free graphs

For a connected graph G = ( V , E ) , an edge set S ⊂ E is a k -restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k -restricted edge connectivity of G , denoted by λ k ( G ) , is defined as the cardinality of a minimum k -restricted edge cut. For U 1 , U 2 ⊂ V ( G ) , denote the set of edges of G with one end in U 1 and the other end in U 2 by [ U 1 , U 2 ] . Define ξ k ( G ) = min { | [ U , V ( G ) ∖ U ] | : U ⊂ V ( G ) , | U | = k ≥ 1 , and the subgraph induced by U is connected}. A graph G is λ k -optimal if λ k ( G ) = ξ k ( G ) . Let k be a positive integer, and let G be a connected triangle-free graph of order n ≥ 2 k . In this paper, we prove that (a) If d ( u ) + d ( v ) ≥ 2 ⌊ n + 2 4 ⌋ + 1 for each pair u , v ∈ V ( G ) such that the distance between u and v is 2 , then G is λ 2 -optimal; (b) If there are at least k common vertices in the neighbor sets of each pair of nonadjacent vertices in G , then G is λ k -optimal.