Sufficient conditions for triangle-free graphs to be optimally restricted edge-connected
Discrete Applied Mathematics
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An edge set S is a k-restricted edge cut of a connected graph G if G-S is no longer connected and every component of G-S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is the cardinality of a minimum k-restricted edge cut. A graph G with λk(G) = ξk(G) is called λk-optimal, where ξk(G) = min{∣U,Ū∣ ∣ U ⊂ V(G),∣U∣ = k and GU is connected}, ∣U,Ū∣ is the number of edges between U and Ū, GU is the subgraph of G induced by U. In this article, we give a sufficient condition for a graph to be λk-optimal: for any integer k ≥ 3, every graph G with girth g ≥ 5, minimum degree δ ≥ k, and diameter D ≤ g - 4 when g is even and D ≤ g - 3 when g is odd is λk-optimal. Furthermore, if δ ≥ 2k - 3, the diameter condition can be relaxed a little to D ≤ g - 3 no matter whether g is even or odd. This generalizes a result of Balbuena et al. [Sufficient conditions for λ′-optimality in graphs with girth g, J Graph Theory 52 (2006) 73–86], and improves a result of Fàbrega and Fiol in a sense [On the extraconnectivity of graphs, Discrete Math 155 (1996) 49–57]. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010