Dense subgraphs in the H-free process

Abstract

The H-free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of H is created, where H is some fixed graph. When H is strictly 2-balanced, we show that for some c, d > 0, with high probability as n → ∞, the final graph of the H-free process contains no subgraphs F on v F ≤ n d vertices with maximum density max J⊆F {e J /v J } ≥ c. This extends and generalizes results of Gerke and Makai for the C 3 -free process.

The Final Size of theC4-Free Process

“…For random graph G(n, i) to have the property that the addition of any new edge creates a copy of H, provided H is strictly 2-balanced (see [12]). It is also known (see [7] and [14]) that sufficiently dense subgraphs are unlikely to appear in the final graph G(M(H)).…”

The diamond‐free process

“…(Similar results for the triangle-free process were obtained by Wolfovitz [14]). Extending results of Gerke and Makai [5] for the triangle-free process, Warnke [10] showed that there are constants c 1 , c 2 depending only on H so that, w.h.p., the graph G M (H) contains no subgraphs on at most n c 1 vertices with density at least c 2 . As a result of our analysis, one can modify Warnke's approach to show the existence of constants c 1 and c 2 so that the same conclusion holds for G M (K − 4 ) ; we leave these details to the interested reader.…”