Degree powers in graphs with a forbidden even cycle

Abstract

Let Cl denote the cycle of length l. For p ≥ 2 and integer k ≥ 1, we prove that the function φ (κ, p, n) = max{ ∑/u∈V(G) dp (u) : G is a graph of order n containing no C2k+2} satisfies φ (κ, p, n) = knp (1 + o (1)). This settles a conjecture of Caro and Yuster. Our proof is based on a new sufficient condition for long paths.