Maximum sparse induced subgraphs of the binomial random graph with given number of edges

The Maximum Number of Edges in a Strongly Multiplicative Graph

The Maximum Number of Complete Subgraphs of Fixed Size in a Graph with Given Maximum Degree

In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs G with n vertices and ∆(G) ≤ r, which has the most complete subgraphs of size t, for t ≥ 3. The conjectured extremal graph is aKr+1 ∪Kb, where n = a(r + 1) + b with 0 ≤ b ≤ r. Gan, Loh, and Sudakov proved the conjecture wh...

Induced subgraphs in sparse random graphs with given degree sequences

Let Gn,d denote the uniformly random d-regular graph on n vertices. For any S ⊂ [n], we obtain estimates of the probability that the subgraph of Gn,d induced by S is a given graph H. The estimate gives an asymptotic formula for any d = o(n1/3), provided that H does not contain almost all the edges of the random graph. The result is further extended to the probability space of random graphs with...

Induced subgraphs in sparse random graphs with given degree sequence

Let Gn,d denote the uniformly random d-regular graph on n vertices. For any S ⊂ [n], we obtain estimates of the probability that the subgraph of Gn,d induced by S is a given graph H. The estimate gives an asymptotic formula for any d = o(n1/3), provided that H does not contain almost all the edges of the random graph. The result is further extended to the probability space of random graphs with...

the maximum number of edges in a strongly multiplicative graph

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