In this paper we show that the graph of k-ary trees, connected by rotations, contains a Hamilton cycle. Our proof is constructive and thus provides a cyclic Gray code for k-ary trees. Furthermore, we identify a basic building block of this graph as the 1-skeleton of the polytopal complex dual to the lower faces of a certain cyclic polytope.

In 2006, Kühn and Osthus showed that if a 3-graph H on n vertices has minimum co-degree at least (1/4 + o(1))n and n is even then H has a loose Hamilton cycle. In this paper, we prove that the minimum co-degree of n/4 suffices. The result is tight.

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n → ∞, provided d ≥ 8.

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This proves a conjecture of Penrose. We also show that in the k-nearest neighbour model, there is a constant κ such that almost every κ-connected graph has a Hamilton cycle.

In this paper, we investigate the maximally parallel attribute of P Systems. Some properties of P Systems are abstracted, which are the filter property and the enumeration property. They are applied to solve the sorting problem and the Hamilton cycle problem, respectively.

We study the comovement of international business cycles in a time-series clustering model with regime switching. extend framework Hamilton and Owyang (2012) to include time-varying transition probabilities determine what drives simultaneous cycle turning points. find four groups, or “clusters,” countries that experience idiosyncratic recessions relative global cycle. In addition, we primary in...

A rainbow subgraph of an edge-coloured graph has all edges of distinct colours. A random d-regular graph with d even, and having edges coloured randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with probability tending to 1 as n→∞, for fixed d ≥ 8.

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbour model, there is a constant κ such that almost every κ-connected graph has a Hamilton cycle.

When the directed graph of an n–by–n matrix A does not contain a Hamilton cycle, we exhibit a formula for detA in terms of sums of products of proper principal minors of A. The set of minors involved depends upon the zero/nonzero pattern of A.

Let K (k) n be the complete k-uniform hypergraph, k ≥ 3, and let ` be an integer such that 1 ≤ ` ≤ k − 1 and k − ` divides n. An `-overlapping Hamilton cycle in K (k) n is a spanning subhypergraph C of K (k) n with n/(k− `) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of adjacent edges in C intersects in precisely ...