The inducibility of complete bipartite graphs

Abstract

For any s 2: 1 and t 2: (n, we prove that among all graphs with n vertices the graph that contains the maximal number of induced copies of K r , Hs for any fixed s 2: 1 and t 2: (n is K(n/2)+a. (n/2)-a for some function a = o(n). We show that this is not valid for t < (n.Analogous results for complete multipartite graphs are also obtained.

The Inducibility of Graphs on Four Vertices

“…This resolved the case of K 2, 2 and its complement. Later in , Brown and Sidorenko proved that if H is a complete bipartite graph, then the graph which maximizes can also be chosen to be complete bipartite, resolving K 1, 3 and its complement. They also gave a generalization to complete multipartite graphs, along with conditions under which the exact graph G is known.…”

“…There remain three 4‐vertex graphs (considering one from each complementary pair) for which the exact value of is unresolved: K 1, 1, 2 , the paw graph (the graph obtained from a triangle by adding a pendant edge) H paw , and the path on four vertices P 4 . Brown and Sidorenko mentioned in that the best construction they know for K 1, 1, 2 is a balanced complete multipartite graph with five parts (a “5‐equipartite graph” by their convention). Making use of the recent theory of flag algebras from and semidefinite programming techniques used in [, , , , ] among other papers, we determine , and , and show that the construction of Brown and Sidorenko is in fact best possible.…”

The maximal number of inducedr-partite subgraphs

“…Though this is not true in general, it does hold if, for a given r, t is not too small. In our final theorem, we come close to identifying the pairs (r, t) for which T,(n) is an extremal graph for every n. In a somewhat weaker form, the result has been proved, independently, by Brown and Sidorenko [4].…”

“…From a recent result of Brown and Sidorenko [4] we know that T,(n) is asymptotically optimal for this missing K~(t) in Theorem 13. It is likely that T,(n) is, in fact, an extremal graph for the missing value of t as well, though judging by the case r = 3, t = 2 (Theorem 9), this may well be rather difficult to prove.…”

“…Among other results, they proved that if r < s and G is a Ks-free graph of order n then f(G, Kr)<_ f(T~_l(n),K,). Also, Brown and Sidorenko [4] proved, log(r + 1) among others, if t > then the Tur~m graph T,(n) is asymptotically flog(1 + l/r) optimal for K,(t). They also showed that if F is any complete multipartite graph with at least two parts then, for any n, there is a complete multipartite graph of order n which contains at least as many induced copies of F as any graph of order n. In particular, f(n, K,(t)) = f(G, K,(t)) for some complet e multipartite graph G of order n. From this one can show that f(n, K,(t)) is as above.…”