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# K1,Delta+1-Free and Komega+1-Free Graphs with Many Cliques

#### Abstract

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 *T*_{ω }(*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

*K*for each size

_{t}*t.*Cutler and Radcliffe proved that among graphs with

*n*vertices and maximum degree at most Δ,

*aK*

_{Δ+1}∪

*K*has the maximum number of cliques, where

_{ b}*n*=

*a*(Δ + 1)+

*b*and 0 ≤

*b*≤ Δ, answering a question of Galvin. Gan, Loh, and Sudakov conjectured that

*aK*

_{Δ+1}∪

*K*also maximizes the number of copies of

_{ b}*K*for each size

_{t}*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

*aK*

_{Δ+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

*K*per vertex. We define

_{t}*f*(Δ,ω) as the supremum of

^{t }*ρ*the number of copies of

_{t },*K*per vertex, among such graphs, and show for fixed

_{t}*t*and ω that

*f*(Δ,ω)= (1+

_{t}*o*(1))

*ρ*(T

_{t}_{ω}(Δ+[Δ / ω–1]). For two infinite families of pairs (Δ,ω), we determine

*f*(Δ,ω) exactly for all

_{t}*t*≥ 3. For another we determine

*f*(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

_{t}*K*per vertex for every size

_{t}*t.*^

#### Subject Area

Mathematics

#### Recommended Citation

Kirsch, Rachel, "K1,Delta+1-Free and Komega+1-Free Graphs with Many Cliques" (2018). *ETD collection for University of Nebraska - Lincoln*. AAI10793912.

https://digitalcommons.unl.edu/dissertations/AAI10793912