Small Subgraphs in Random Distance 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).…”

“…This work is focused on thresholds for the property of cycle containment in G p (n, r, s). In [18] Burkin found the threshold for containment in G p (n, r, s) of a subgraph isomorphic to a fixed graph (under certain assumptions). Applied to a simple cycle C t of a fixed length t, his theorem can be stated as follows.…”

“…Note that if t = const, then there is no sharp threshold, which follows from the fact that in this case for p = c • n −(r−s)−s/t , where c > 0 is some constant, the number of cycles has asymptotically Poisson distribution [18].…”

“…The proof of Theorem 1 provided in [18] by Burkin is also based on the methods of the first and the second moments. Let X and c t be the numbers of copies of C t in G p (n, r, s) and in G(n, r, s) respectively.…”