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

In this article, 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 Δ(G) ≤ r, which has the most complete subgraphs of size t, for t≥3. The conjectured extremal graph is aK_r+1 ∪ K_b, where n = a(r + 1) + b with 0 ≤ b ≤ r. Gan et al. (Combin Probab Comput 24(3) (2015), 521–527) proved the conjecture when a ≤ 1, and also reduced the general conjecture to the case t = 3. We prove the conjecture for r ≤ 6 and also establish a weaker form of the conjecture for all r.