Degree powers in graphs with a forbidden forest

Abstract

Given a positive integer p and a graph G with degree sequence d1,…,dn, we define ep(G)=∑i=1ndip. Caro and Yuster introduced a Turán-type problem for ep(G): Given a positive integer p and a graph H, determine the function exp(n,H), which is the maximum value of ep(G) taken over all graphs G on n vertices that do not contain H as a subgraph. Clearly, ex1(n,H)=2ex(n,H), where ex(n,H) denotes the classical Turán number. Caro and Yuster determined the function exp(n,Pℓ) for sufficiently large n, where p≥2 and Pℓ denotes the path on ℓ vertices. In this paper, we generalise this result and determine exp(n,F) for sufficiently large n, where p≥2 and F is a linear forest. We also determine exp(n,S), where S is a star forest; and exp(n,B), where B is a broom graph with diameter at most six.