The Super-Connectivity of Kneser Graphs

Gülnaz Boruzanli Ekinci 1  and John Baptist Gauci 2
  • 1 Ekinci Department of Mathematics, Faculty of Science Ege University, Bornova,, Izmir, Turkey
  • 2 Department of Mathematics, Faculty of Science University of Malta,, Msida, Malta

Abstract

A vertex cut of a connected graph G is a set of vertices whose deletion disconnects G. A connected graph G is super-connected if the deletion of every minimum vertex cut of G isolates a vertex. The super-connectivity is the size of the smallest vertex cut of G such that each resultant component does not have an isolated vertex. The Kneser graph KG(n, k) is the graph whose vertices are the k-subsets of {1, 2, . . . , n} and two vertices are adjacent if the k-subsets are disjoint. We use Baranyai’s Theorem on the decompositions of complete hypergraphs to show that the Kneser graph KG are super-connected when n ≥ 5 and that their super-connectivity is n ( n/2) − 6 when n ≥ 6.

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