On the diameter of generalized Kneser graphs

Abstract Let r , k be positive integers, s ( r ) , a nonnegative integer, and n = 2 r - s + k . The set of r-subsets of [ n ] = { 1 , 2 , … , n } is denoted by [ n ] r . The generalized Kneser graph K ( n , r , s ) is the graph whose vertex-set is [ n ] r where two r-subsets A and B are joined by an edge if | A ∩ B | ⩽ s . This note determines the diameter of generalized Kneser graphs. More precisely, the diameter of K ( n , r , s ) is equal to ⌈ r - s - 1 s + k ⌉ + 1 , which generalizes a result of Valencia–Pabon and Vera [On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383–385].