In this note, nonlinear time delay systems with homogeneous right-hand sides of degree greater than one and a constant delay are studied. Under an assumption that the corresponding delay free system is asymptotically stable, two criteria for asymptotic stability of the trivial solution based on the Lyapunov-Krasovskii functionals are presented. The first of them is a converse of the well-known Krasovskii theorem under the mentioned assumption whereas in the second one the functional is required to admit a positive definite lower bound on the set of functions satisfying a special condition of Razumikhin type only. The Lyapunov-Krasovskii functionals, which satisfy the conditions of these criteria and are universal within the class of systems under consideration, are constructed. These functionals are based on the Lyapunov functions suitable for analysis of the corresponding delay free systems. They are applied to estimation of the attraction region of solutions of homogeneous systems. Copyright (C) 2019. The Authors. Published by Elsevier Ltd. All rights reserved.