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K1,Δ+1-Free and Kω+1-Free Graphs with Many Cliques
The problem of maximizing the number of cliques of size t has been studied within several classes of graphs. Zykov showed that among graphs on n vertices with clique number ω( G) ≤ ω, the Turán graph Tω (n) maximizes the number of copies of K t for each size t. A corollary of the Kruskal-Katona theorem shows that among graphs on m edges, the colex graph C(m) maximizes the number of copies of Kt for each size t. Cutler and Radcliffe proved that among graphs with n vertices and maximum degree at most Δ, aKΔ+1 ∪ K b has the maximum number of cliques, where n = a(Δ + 1)+ b and 0 ≤ b ≤ Δ, answering a question of Galvin. Gan, Loh, and Sudakov conjectured that aKΔ+1 ∪ K b also maximizes the number of copies of Kt for each size t ≥ 3. They proved this conjecture for a = 1, and Cutler and Radcliffe proved it for Δ ≤ 6. We solve two related problems. First, we investigate a variant of the Gan-Loh-Sudakov conjecture, where we fix the number of edges instead of the number of vertices. We prove that aKΔ+1 ∪ C(b) maximizes the number of triangles among graphs with m edges and any fixed maximum degree at most Δ ≤ 8, where m = a(Δ+1 / 2 ) + b and 0 ≤ b < (Δ+1 / 2). Second, we combine the restrictions on maximum degree and clique number and investigate which graphs with Δ(G) ≤ Δ and ω(G) ≤ ω maximize the number of copies of Kt per vertex. We define ft (Δ,ω) as the supremum of ρt , the number of copies of Kt per vertex, among such graphs, and show for fixed t and ω that ft(Δ,ω)= (1+o(1)) ρt(Tω(Δ+[Δ / ω–1]). For two infinite families of pairs (Δ,ω), we determine ft(Δ,ω) exactly for all t ≥ 3. For another we determine ft(Δ,ω) exactly for the two largest possible clique sizes. Finally, we demonstrate that not every pair (Δ,ω) has an extremal graph that simultaneously maximizes the number of copies of Kt per vertex for every size t.
Kirsch, Rachel, "K1,Δ+1-Free and Kω+1-Free Graphs with Many Cliques" (2018). ETD collection for University of Nebraska - Lincoln. AAI10793912.