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Abstract For a given pair of two graphs (F , H) , let R (F , H) be the smallest positive integer r such that for any graph G of order r , either G contains F as a subgraph or the complement of G contains H as a subgraph. Baskoro, Broersma and Surahmat (2005) conjectured that R (F ℓ , K n) = 2 ℓ (n − 1) + 1 for ℓ ≥ n ≥ 3 , where F ℓ is the join K 1 + ℓ K 2 of K 1 and ℓ K 2. In this paper, we prove that this conjecture is true for the case n = 6. [ABSTRACT FROM AUTHOR]

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 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) = 2 ex (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. [ABSTRACT FROM AUTHOR]