Degree powers in graphs with a forbidden forest (original) (raw)

2019, Discrete Mathematics

Given a positive integer p and a graph G with degree sequence d 1 ,. .. , d n , we define e p (G) = n i=1 d p i. Caro and Yuster introduced a Turán-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) = 2ex(n, H), where ex(n, H) denotes the classical Turán 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.