On weakly Turán-good graphs

Abstract

Given graphs H and F with χ(H) < χ(F ), we say that H is weakly F -Turán-good if among n-vertex F -free graphs, a (χ(F ) − 1)-partite graph contains the most copies of H. Let H be a bipartite graph that contains a complete bipartite subgraph K such that each vertex of H is adjacent to a vertex of K. We show that H is K 3 -Turán-good, improving a very recent asymptotic bound due to Grzesik, Győri, Salia and Tompkins. They also showed that for any r there exists graphs that are not weakly K r -Turán-good. We show that for any F there exists graphs that are not weakly F -Turán-good. We also show examples of graphs that are weakly F -Turán-good but not K 3 -Turán-good for every k > 1, in the case F is 3-chromatic, triangle-free and can be obtained from a bipartite graph by adding a vertex.

Some exact results for generalized Turán problems

“…The odd girth og(G) of a graph G is the shortest odd cycle in it. Its connection to Turán-goodness was studied in [12]. It was shown there that G is the subgraph of the p-blow-up of the cycle of length og(G) for some p, i.e., of the graph obtained the following way: we replace each vertex v of G with p vertices v 1 , .…”

Every graph is eventually Turán-good

“…This conjecture is known to hold for stars [4], more generally for complete multipartite graphs [11], for paths [10] and for the 5-cycle [16]. In this paper we prove the conjecture holds with r 0 = 300v(H) 9 .…”

“…Here we present an informal outline of the proof, which will be given in Section 3. We can assume that n ≥ r + 1 ≥ 300v(H) 9 , and so n is much larger than v(H), as otherwise the theorem trivially holds. Thus almost every ϕ :…”

“…Then H is K r+1 -Turán-good. Theorem 1.2 follows from a more technical result (Theorem 1.3 below, which also implies that for r ≥ 300v(H) 9 Turán graphs always maximize the number of copies of H among K r+1 -free graphs on any given number of vertices, i.e., the requirement that the number of vertices is large compared to r is unnecessary.…”

“…In [16] it is conjectured that Theorem 1.2 should hold with r 0 (H) = v(H) + 1. Counterexamples are known for all r ≤ χ(H) + 1, but determining the exact value of r 0 (H) remains open, see [9,12,15,16]. We did not attempt to optimize our bound, beyond ensuring that it is polynomial in v(H), and prioritized simplicity of presentation.…”