The maximal number of inducedr-partite subgraphs

Abstract

Abstract. In this paper, we extend earlier results concerning the maximal number of induced complete r-partite graphs in a graph G of order n. In particular, we show that if t > 1 + log r, then the maximal number of induced K~(t)'s is achieved in the case of the Tur~n graph T~(n), and that this bound on t is essentially best possible.

The inducibility of blow-up graphs

“…Counting induced copies. As mentioned in the introduction, Bollobás et al [BEHJ95] asked the following question.…”

“…Note that K (h) r , the h-blow-up of the complete graph on r vertices, is the complete r-partite graph where each part has exactly h vertices. Bollobás, Egawa, Harris and Jin [BEHJ95] showed that for sufficiently large h, the maximal graphs for K (h) r are blow-ups of K r . They asked for which graphs H does there exist a constant h such that for sufficiently large n, the graph on n vertices that contains the maximum number of induced copies of H (h) is a blow-up of H?…”

Turán $H$-Densities for 3-Graphs

“…Then Hirst showed that π K 1,1,2 (∅) = 72/125, π paw (∅) = 3/8, with extremal configurations a balanced blow-up of K 5 and the complement of a balanced blow-up of ( [4], {12, 34}) respectively. Sperfeld proved π C 3 (∅) = 1/4, π K 2 ⊔E 1 (∅) = 3/4, with extremal configurations a random tournament on n vertices and the disjoint union of two tournaments on n/2 vertices respectively.…”

A Note on the Inducibility of $$4$$ 4 -Vertex Graphs

“…In general, by passing from G to its complement G C one can easily obtain I(H) = I(H C ). The inducibility of complete partite graphs has been considered in the literature [26,10,2,3,1]. For example, the inducibility of the complete bipartite graph K t,t , is I(K t,t ) = 2t t /4 t ≈ 1/ √ πt, as attained by larger bipartite graphs.…”

“…There are eleven isomorphism types of 4-vertex graphs, the inducibilities of ten of which are known, and summarized in Table 1, an updated version of Exoo's [10]. Some of these numbers follow from general results concerning complete bipartite graphs [26,2,3,1]. For others, various extremal constructions were found, and their optimality was proved using flag algebra [18].…”