Subdivisions with Distance Constraints in Large Graphs

In this dissertation we are concerned with sharp degree conditions that guarantee the existence of certain types of subdivisions in large graphs. Of particular interest are subdivisions with a certain number of arbitrarily specified vertices and with prescribed path lengths. Our non-standard approach makes heavy use of the Regularity Lemma (Szemerédi, 1978), the Blow-Up Lemma (Komlós, Sárkózy, and Szemerédi, 1994), and the minimum degree panconnectivity criterion (Williamson, 1977).Sharp minimum degree criteria for a graph G to be H-linked have recently been discovered. We define (H,w,d)-linkage, a condition stronger than H-linkage, by including a weighting function w consisting of required lengths for each edge-path of a desired H-subdivision. We establish sharp minimum degree criteria for a large graph G to be (H,w,d)-linked for all nonnegative d. We similarly define the weaker condition (H,S,w,d)-semi-linkage, where S denotes the set of vertices of H whose corresponding vertices in an H-subdivision are arbitrarily specified. We prove similar sharp minimum degree criteria for a large graph G to be (H,S,w,d)-semi-linked for all nonnegativeWe also examine path coverings in large graphs, which could be seen as a special case of (H,S,w)-semi-linkage. In 2000, Enomoto and Ota conjectured that a graph G of order n with degree sum σ2(G) satisfying σ2(G) \u3e n + k - 2 may be partitioned into k paths, each of prescribed order and with a specified starting vertex. We prove the Enomoto-Ota Conjecture for graphs of sufficiently large order

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

Color-Critical Graphs Have Logarithmic Circumference

A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100log k), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed 2(k-1)log n/log(k-2). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954

Factorization of Rational Curves in the Study Quadric and Revolute Linkages

Given a generic rational curve $C$ in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly $C$. Our construction is based on the factorization of polynomials over dual quaternions. Low degree examples include the Bennett mechanisms and contain new types of overconstrained 6R-chains as sub-mechanisms.Comment: Changed arxiv abstract, corrected some type

Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs

Precise Partitions Of Large Graphs

First by using an easy application of the Regularity Lemma, we extend some known results about cycles of many lengths to include a specified edge on the cycles. The results in this chapter will help us in rest of this thesis. In 2000, Enomoto and Ota posed a conjecture on the existence of path decomposition of graphs with fixed start vertices and fixed lengths. We prove this conjecture when |G| is large. Our proof uses the Regularity Lemma along with several extremal lemmas, concluding with an absorbing argument to retrieve misbehaving vertices. Furthermore, sharp minimum degree and degree sum conditions are proven for the existance of a Hamiltonian cycle passing through specified vertices with prescribed distances between them in large graphs. Finally, we prove a sharp connectivity and degree sum condition for the existence of a subdivision of a multigraph in which some of the vertices are specified and the distance between each pair of vertices in the subdivision is prescribed (within one)