Bounding ℓ-edge-connectivity in edge-connectivity

Abstract

For a connected graph G , let κ ′ ( G ) be the edge-connectivity of G . The ℓ - edge-connectivity κ ℓ ′ ( G ) of G with order n ≥ ℓ is the minimum number of edges that are required to be deleted from G to produce a graph with at least ℓ components. It has been observed that while both κ ′ ( G ) and κ ℓ ′ ( G ) are related edge connectivity measures. In general, κ ℓ ′ ( G ) cannot be upper bounded by a function of κ ′ ( G ) . Let κ ¯ ′ ( G ) = max { κ ′ ( H ) : H ⊆ G } be the maximum subgraph edge-connectivity of G . We prove that for integers k ′ , k and ℓ with k ′ ≥ k ≥ 1 and ℓ ≥ 2 , each of the following holds. (i) sup { κ ℓ ′ ( G ) : κ ′ ( G ) = k , κ ¯ ′ ( G ) = k ′ } = k + ( ℓ − 2 ) k ′ . (ii) inf { κ ℓ ′ ( G ) : κ ′ ( G ) = k , κ ¯ ′ ( G ) = k ′ } = k ℓ 2 .