Small subgraphs and their extensions in a random distance graph

Large cycles in generalized Johnson graphs

“…Unfortunately, there is no established term for graphs $$G(n,r,s)$$. In literature they appear as generalized Johnson graphs [1, 7, 33]; uniform subset graphs [9, 11, 41] and distance graphs [5, 6, 36, 45]. The family of $$G(n,r,s)$$ graphs was initially (to the best of our knowledge) considered in [9], where they are called “ uniform subset graphs.…”

Large cycles in generalized Johnson graphs

“…For integers n, r, s such that 0 ≤ s < r < n, a simple graph G(n, r, s) with the set of vertices Unfortunately, there is no established term for graphs G(n, r, s). In literature they appear as generalized Johnson graphs [1,7,33]; uniform subset graphs [9,11,41] and distance graphs [5,6,36,45]. The family of G(n, r, s) graphs was initially (to the best of our knowledge) considered in [9], where they are called "uniform subset graphs".…”

Large cycles in random generalized Johnson graphs

“…Particularly well studied is the property of subgraph containment [1,7,8,9,10,11]. In more recent works some results concerning the asymptotic properties of random Kneser graphs [12,13,14] and of G p (n, r, s) [15,16,17,18,19,20] can be found. This work is focused on thresholds for the property of cycle containment in G p (n, r, s).…”