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....

In this paper we present a min-max theorem for a factorization problem in directed graphs. This extends the Berge-Tutte formula on matchings as well as formulas for the maximum even factor in weakly symmetric directed graphs and a factorization problem in undirected graphs. We also prove an extension to the structural theorem of Gallai and Edmonds about a canonical set attaining minimum in the ...

In [4], we have proved the necessary condition for sigraphs on union of k copies cycles of length three with one vertex in common denoted by C 3 to be graceful and sufficiency part of the theorem was discussed in certain special cases when all rim edges are negative and form a maximum matching. In this paper, we discuss the sufficiency part of the theorem on gracefulness of sigraphs on C 3 part...

the most important factor in donor-recipient matching for hematopoietic stem cell transplantation is their hla typing. for many patients, we can not find a full-matched donor. in high risk patients, one locus  mismatched related donors are suitable alternatives. (1, 2) since the matching of patients and their siblings usually is based on low resolution hla typing, the matching of one locus mism...

We use a simple matrix splitting technique to give an elementary new proof of the Lidskii-Mirsky-Wielandt Theorem and to obtain a multiplicative analog of the Lidskii-Mirsky-Wielandt Theorem, which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of Hermitian matrices and singular values of general matrices. We apply our bound to obtain numerous bou...

In this paper we study the (classical) problem of reconstructing a continuous-time signal from its samples. We adopt a system-theoretical viewpoint of this problem. Namely, we assume that the continuous-time signal is modeled as the output of a linear time-invariant system and then formulate the reconstruction problem as the (either L or L) optimal model-matching problem in the lifted domain. T...

Abstract We demonstrate for the €rst time that Liquid Haskell, a re€nement type checker for Haskell programs, can be used for arbitrary theorem proving by verifying a monoidal string matching algorithm implemented in Haskell. We use re€nement types to specify correctness properties, Haskell terms to express proofs of these properties, and Liquid Haskell to check the proofs. We evaluate Liquid H...

the aim of this work is to describe the qualitative behavior of the solution set of a givensystem of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. in order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. this is done by the extension of ...

Local search for combinatorial optimization problems is becoming a dominant algorithmic paradigm, with several papers using it to resolve long-standing open problems. In this paper, we prove the following ‘4-local’ version of Hall’s theorem for planar graphs: given a bipartite planar graph G = (B,R,E) such that |N(B′)| ≥ |B′| for all |B′| ≤ 4, there exists a matching of size at least |B| 4 in G...

This paper describes a prototype of a programmable interactive theorem-proving system. The main new feature of this system is that it supports the construction and manipulation of tree-structured proofs that can contain both metavariables and derived rules that are computed by tactic programs. The proof structure encapsulates the top-down reenement process of proof construction typical of most ...