Degree powers in graphs with a forbidden forest

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