Some exact results of the generalized Turán numbers for paths

Abstract

For graphs H and F with chromatic number χ(F)=k, we call H strictly F-Turán-good (or (H,F) strictly Turán-good) if the Turán graph Tk−1(n) is the unique F-free graph on n vertices containing the largest number of copies of H when n is large enough. Let F be a graph with chromatic number χ(F)≥3 and a color-critical edge and let Pℓ be a path with ℓ vertices. Gerbner and Palmer (2020) showed that (P3,F) is strictly Turán good if χ(F)≥4 and they conjectured that (a) this result is true when χ(F)=3, and, moreover, (b) (Pℓ,Kk) is Turán-good for every pair of integers ℓ and k. In the present paper, we show that (H,F) is strictly Turán-good when H is a bipartite graph with matching number ν(H)=⌊|V(H)|2⌋ and χ(F)=3, as a corollary, this result confirms the conjecture (a); we also prove that (Pℓ,F) is strictly Turán-good for 2≤ℓ≤6 and χ(F)≥4, this also confirms the conjecture (b) for 2≤ℓ≤6 and k≥4.