This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemerédi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs....

We study M-alternating Hamilton paths, and M-alternating Hamilton cycles in a simple connected graph G on ν vertices with a perfect matchingM . Let G be a bipartite graph, we prove that if for any two vertices x and y in different parts of G, d(x)+d(y) ≥ ν/2+2, then G has an M-alternating Hamilton cycle. For general graphs, a condition for the existence of an M-alternating Hamilton path startin...

A k-factor of a graph is a k-regular spanning subgraph. A Hamilton cycle is a connected 2-factor. A graph G is said to be primitive if it contains no k-factor with 1 ≤ k < ∆(G). A Hamilton decomposition of a graph G is a partition of the edges of G into sets, each of which induces a Hamilton cycle. In this paper, by using the amalgamation technique, we find necessary and sufficient conditions f...

A graph is cubic if each of its vertex is of degree 3 and it is hamiltonian if it contains a cycle passing through all its vertices. It is known that if a cubic graph is hamiltonian, then it has at least three Hamilton cycles. This paper is about those works done concerning the number of Hamilton cycles in cubic graphs and related problems.

The symmetric group is generated by σ = (1 2 ··· n) and τ = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n. Dedicated to Herb Wilf (1931 – 2012).

In this paper we extend general grid graphs to the grid graphs consist of polygons tiling on a plane, named polygonal grid graphs. With a cycle basis satisfied polygons tiling, we study the cyclic structure of Hamilton graphs. A Hamilton cycle can be expressed as a symmetric difference of a subset of cycles in the basis. From the combinatorial relations of vertices in the subset of cycles in th...

An n-bit binary Gray code is an enumeration of all n-bit binary strings so that successive elements differ in exactly one bit position; equivalently, a hamilton path in the Hasse diagram of Bn (the partially ordered set of subsets of an n-element set, ordered by inclusion.) We construct, for each n, a hamilton path in Bn with the following additional property: edges between levels i− 1 and i of...

It is well known that every bipartite graph with vertex classes of size nwhose minimum degree is at least n/2 contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac’s theorem on Hamilton cycles for 3-uniform hypergraphs: We say that a 3-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices su...

A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G. A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled by Kühn and Osthus [15], who proved more generally that every r-regular n-vertex o...