The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions
of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).
Authors:R. Faudree, R. Gould, M. Jacobson, L. Lesniak, and T. Lindquester
It is known that if a 2-connected graphG of sufficiently large ordern satisfies the property that the union of the neighborhoods of each pair of vertices has cardinality at leastn/2, thenG is hamiltonian. In this paper, we obtain a similar generalization of Dirac’s Theorem forK(1,3)-free graphs. In particular, we show that ifG is a 2-connectedK(1,3)-free graph of ordern with the cardinality of the union of the neighborhoods of each pair of vertices at least (n+1)/3, thenG is hamiltonian. We also investigate several other related properties inK(1,3)-free graphs such as traceability, hamiltonian-connectedness, and pancyclicity.