It is well known that not all 3-connected planar graphs are hamiltonian. Whitney [10] proved that every triangulation of the sphere with no separating triangles is hamiltonian. Tutte [9] proved that every 4-connected planar graph has a Hamilton cycle. Extending Tutte's technique, Thomassen [8] proved that every 4-connected planar graph is in fact Hamilton connected. (A small omission in [8] was...

The notorious middle two levels problem is to nd a Hamilton cycle in the middle two levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known result, due to Moews and Reid 11] in 1990, was that M 2k+1 is Hamiltonian for all positive k through k = 11. We show that if a Hamilton path between two ...

In previous papers the authors showed that almost all d-regular graphs for d ≥ 3 are hamiltonian. In the present paper this result is generalized so that a set of j oriented root edges have been randomly specified for the cycle to contain. The Hamilton cycle must be orientable to agree with all of the orientations on the j root edges. It is shown that the requisite Hamilton cycle almost surely ...

A classic result of G. A. Dirac in graph theory asserts that every n-vertex graph (n ≥ 3) with minimum degree at least n/2 contains a spanning (so-called Hamilton) cycle. G. Y. Katona and H. A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutiv...

The notorious middle two levels problem is to find a Hamilton cycle in the middle two levels, M2k+1, of the Hasse diagram of B2k+1 (the partially ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known result, due to Moews and Reid [11] in 1990, was that M2k+1 is Hamiltonian for all positive k through k = 11. We show that if a Hamilton path between two d...

Let G = Gn(a1, a2, ..., ak) denote a directed circulant graph of order n with k pairwise distinct jumps [1]. Antidirected Hamilton cycle in G is a cycle of n arcs that does not contain induced directed path P2. Let C 1 G, C 2 G, ..., C k G be pairwise arc-disjoint antidirected Hamilton cycles in G, each composed of exactly two distinct jumps. We give the necessary and sufficient conditions for ...

We will present a new method, which enables us to find threshold functions for many properties in random intersection graphs. This method will be used to establish sharp threshold functions in random intersection graphs for k–connectivity, perfect matching containment and Hamilton cycle containment. keywords: random intersection graph, threshold functions, connectivity, Hamilton cycle, perfect ...

In the random k-uniform hypergraph Hn,p;k of order n each possible k-tuple appears independently with probability p. A loose Hamilton cycle is a cycle of order n in which every pair of adjacent edges intersects in a single vertex. We prove that if pn/ logn tends to infinity with n then lim n→∞ 2(k−1)|n Pr(Hn,p;k contains a loose Hamilton cycle) = 1.