The super-connectivity of Kneser graphs

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(n, 2) are super-connected when n ≥ 5 and that their super-connectivity is n 2 − 6 when n ≥ 6.

The p-restricted edge-connectivity of Kneser graphs

“…Concerning the connectedness of Kneser graphs the following results were obtained in [7] . Note that K ( n , k ) is connected whenever n ≥ 2 k + 1 , since it has a finite diameter (see again [25] ).…”

The general position problem on Kneser graphs and on some graph operations

“…If n and k are positive integers with n ≥ k, then the Kneser graph K(n, k) has as vertices all the k-element subsets of the set [n], vertices being adjacent if the corresponding sets are disjoint. For more on Kneser graph see [2,3,15,19].…”

The super-connectivity of odd graphs and of their kronecker double cover

“…It has long been conjectured that the Kneser graphs and the bipartite Kneser graphs have a Hamiltonian cycle, apart from one notorious exception, namely the Petersen graph KG (5,2). The sparsest among these graphs (that is, those graphs for which the ratio of the number of edges to the number of vertices is smallest) are respectively KG(2k + 1, k) and H(2k + 1, k).…”

“…In this case, κ is strictly greater than the connectivity κ = κ(G) of G; otherwise κ = κ. Some examples of graph classes which have been analysed for their super-connectivity are circulant graphs [3], products of various graphs (see [7,8,12,19], and the references therein), hypercubes [13,24,25], generalized Petersen graphs [4], Johnson graphs [5] and Kneser graphs [1,6].…”