On local connectivity of graphs

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The local connectivityκ (u, v) of two vertices u and v in a graph G is the maximum number of internally disjoint u-v paths in G, and the connectivity of G is defined as κ (G) = min {κ (u, v) | u, v ∈ V (G)}. Clearly, κ (u, v) ≤ min {d (u), d (v)} for all pairs u and v of vertices in G. Let δ (G) be the minimum degree of G. We call a graph G maximally connected when κ (G) = δ (G) and maximally locally connected when κ (u, v) = min {d (u), d (v)} for all pairs u and v of vertices in G. In 1993, Topp and Volkmann [J. Topp, L. Volkmann, Sufficient conditions for equality of connectivity and minimum degree of a graph, J. Graph Theory 17 (1993) 695-700] proved that a p-partite graph of order n (G) is maximally connected when n (G) ≤ δ (G) {dot operator} frac(2 p - 1, 2 p - 3) . As an extension of this result, we will show in this work that these conditions even guarantee that G is maximally locally connected. © 2007 Elsevier Ltd. All rights reserved.




Volkmann, L. (2008). On local connectivity of graphs. Applied Mathematics Letters, 21(1), 63–66. https://doi.org/10.1016/j.aml.2006.12.014

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