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.
The convexity lattices, introduced by Bennett and Birkhoff, generalize the lattices of convex sets. We present three forms
of Parallel Axiom in such lattices and define Euclidean and two classes of non-Euclidean lattices via the number of parallel
lines through a point. The paper deals with these three classes of lattices.