Degree powers of graphs without [formula omitted

Abstract

Let eq(H)=∑i=1ndiq, where (d1,…,dn) is the degree sequence of graph H and q≥1 is an integer. A Turán-type problem of eq(H) is considered by Caro and Yuster: Given a graph Γ, determine the functionexq(n,Γ)=max{eq(H)|HisaΓ-freegraphofordern}.Clearly, ex1(n,Γ) is two times of the classical Turán number of Γ. Let ℓ≥4, s≥0 and Pℓ=v1v2⋯vℓ be a path of order ℓ, and let Bℓ,s be the graph obtained from Pℓ by adding s vertices u1,…,us that are adjacent to the vertex vℓ−1. The graph Bℓ,s is known as a broom. Caro and Yuster determined the value of exq(n,B4,s) when q≥2 and n is sufficiently large. Lan et al. determined the value of exq(n,Bℓ,s) when q≥2, 5≤ℓ≤7 and n is sufficiently large. In this paper, we determine the value of exq(n,B8,s) when q≥2 and n is sufficiently large. This is a solution to a conjecture due to Lan et al. for the case ℓ=8.