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 develop Hamilton–Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton–Jacobi theor...

In this paper, we settle Alspach’s problem in the case of Hamilton cycles and 5cycles; that is, we show that for all odd integers n ≥ 5 and all nonnegative integers h and t with hn + 5t = n(n − 1)/2, the complete graph Kn decomposes into h Hamilton cycles and t 5-cycles and for all even integers n ≥ 6 and all nonnegative integers h and t with hn+5t = n(n−2)/2, the complete graph Kn decomposes i...

In this paper, we introduce two new algorithm to find a Hamilton Circuit in a graph G = (V, E). One algorithm is use a multistage graph as a special NFAs to find all Hamilton Circuit in exponential time; while another is use O(|V|) variant multistage graph as a special state set NFAs to find a fuzzy data of all Hamilton Circuit in polynomial time. The fuzzy data of the data contain those data, ...

The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree ∆ satisfies some basic expansion properties and contains a family of (1−o(1))∆/2 edge disjoint Hamilton cycles, then there also exists a covering of its...

A. D. Davis, G. D. Wells, B. Falk, and M. D. Noseworthy Medical Physics and Applied Radiation Sciences, McMaster University, Hamilton, Ontario, Canada, Medical Physics and Applied Radiation Sciences, St. Joseph's Healthcare, Hamilton, Ontario, Canada, Department of Anesthesia, University of Toronto, Toronto, Ontario, Canada, Physiology and Experimental Medicine, The Hospital for Sick Children, ...

Let G(n, r, s) denote a uniformly random r-regular s-uniform hypergraph onn vertices, where s is a fixed constant and r = r(n) may grow with n. An `-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlapin precisely ` vertices, and 1-overlapping Hamilton cycles are called loose Hamiltoncycles.When r, s ≥ 3 are fixed integers, we establish a thre...

A cycle in a graph G is called a hamilton cycle if it contains every vertex of G. A l-factor of a graph G is a subgraph H of G with the same vertex set as G, such that each vertex of H has degree one. Ringel [S] has generalized the idea of a hamilton cycle to two dimensions. He showed that if n is odd the set of squares in the n-dimensional cube Q,, can be partitioned into subsets such that eac...

We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e. G contains a set of r−o(r) edge-disjoint Hamilton cycles. Here G is a robust outexpander if for every set S which is not too small and not too large, the ‘robust’ outneighbourhood of S is a little larger than S. T...

Deep vein thrombosis (DVT) is a condition, in which an obstruction of blood flow in veins is found due to the formation of blood clots in the deep veins. An objective diagnosis of DVT is based on ultrasonography examination. However, in poor countries, including Indonesia, many hospitals still do not have the facilities of ultrasonography, less vascular specialists, or D-dimer examination and t...