Degree Powers in Graphs with Forbidden Subgraphs

For every real p > 0 and simple graph G, set f (p, G) = uV (G) dp (u) , and let (r, p, n) be the maximum of f (p, G) taken over all Kr+1-free graphs G of order n. We prove that, if 0 < p < r, then (r, p, n) = f (p, Tr (n)) , where Tr (n) is the r-partite Turan graph of order n. For every p r + 2r and n large, we show that (p, n, r) > (1 + ) f (p, Tr (n)) for some = (r) > 0. Our results settle two conjectures of Caro and Yuster.

• Type: Journal
• Year: 2004
• Where: COMBINATORICS
• Authors: Béla Bollobás, Vladimir Nikiforov
• Keywords: COMBINATORICS 2004, Kr+1-free Graphs, R-partite Turan Graph, Simple Graph