Paths are Turán-good

Abstract

We show that among K k+1 -free n-vertex graphs, the Turán graph contains the most copies of any path.

Every graph is eventually Turán-good

“…Gerbner and Palmer [11,Conjecture 20] conjectured that for every graph H there exists r 0 = r 0 (H) such that H is K r+1 -Turán-good for every r ≥ r 0 (H). 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 .…”