Existence of Dλ-cycles and Dλ-paths

A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way

Lagrangian submanifolds and Lefschetz pencils

Given a Lagrangian submanifold in a symplectic manifold and a Morse function on the submanifold, we show that there is an isotopic Morse function and a symplectic Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From this construction we define a sequence of symplectic invariants classifying the isotopy classes of Lagrangian spheres in a symplectic 4-manifold.Comment: 40 pages, 1 figur

Existence of spanning and dominating trails and circuits

Let T be a trail of a graph G. T is a spanning trail (S-trail) if T contains all vertices of G. T is a dominating trail (D-trail) if every edge of G is incident with at least one vertex of T. A circuit is a nontrivial closed trail. Sufficient conditions involving lower bounds on the degree-sum of vertices or edges are derived for graphs to have an S-trail, S-circuit, D-trail, or D-circuit. Thereby a result of Brualdi and Shanny and one mentioned by Lesniak-Foster and Williamson are improved

A generalization of Ore's Theorem involving neighborhood unions

AbstractLet G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-connected and |N(u)∪N(v)|⩾12n for every pair of nonadjacent vertices u,v, then either G is hamiltonian, or G is the Petersen graph, or G belongs to one of three families of exceptional graphs of connectivity 2