List-coloring and sum-list-coloring problems on graphs

Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained. A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs. We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure. Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles

Universal graphs with forbidden subgraphs and algebraic closure

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated''algebraic closure'' operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases

The Tutte Polynomial Formula for the Class of Twisted Wheel Graphs

The 20th century work of William T. Tutte developed a graph polynomial that is modernly known as the Tutte polynomial. Graph polynomials, such as the Tutte polynomial, the chromatic polynomial, and the Jones polynomial, are at the heart of combinatorical and algebraic graph theory and can be used as tools with which to study graph invariants. Graph invariants, such as order, degree, size, and connectivity which are defined in Section 2, are graph properties preserved under all isomorphisms of a graph. Thus any graph polynomial is not dependent upon a particular labeling or drawing but presents relevant information about the abstract structure of the graph. The Tutte polynomial is the most general graph polynomial that satisfies the recurrence relationship of deletion and contraction. Deletion and contraction, collectively known as the reduction operations and defined in Section 2, are two important actions that con be performed upon a graph in order to aid in the computation of the graph polynomial of interest. The deletion and contraction recurrence relationship states that for every edge e of a graph G, the polynomial of G equals the sum of the polynomial of G delete e and the polynomial of G contract e. Even with the help of these reduction operations, the Tutte polynomial of a graph can be hard to compute with only pen and paper, leading to occasions in which researchers approach the task of developing a formula for the Tutte polynomial of some family of graphs; i.e. a collection of graphs that adhere to common properties. In this thesis, we review the work necessary to compute the Tutte polynomial of the class of fan graphs and the class of wheel graphs and then add to the collection known formulas by computing the formula for the Tutte polynomial of the class of twisted wheel graphs

Navigating time : a portfolio of compositions

This practice-based composition PhD consists of a portfolio of creative work and a supporting commentary. The portfolio illustrates the main themes of my compositional thinking between the years 2005 and 2008 and in particular focuses on my approach to the role of time and therefore form in music. In the works, musical time is considered from a variety of perspectives, but with one essential goal: to explore form as an energised but essentially frozen surface, through and around which performers and, at times, the audience are given the ability to 'navigate'. In essence this approach calls for a personal synthesis between linear and multilinear (or 'modular') musical thinking and lends itself not just to scores for conventional concert instruments, but extends to encompass elements of 'tumtablism', interactive media and the creation of site-specific sound installations. The seven works presented in the portfolio are: Vent/Glacier (2008) for 'prepared' tuba and electronics. Entanglement Laws (2006-2007) for two saxophones, alto trombone, guitar, 4 keyboards and percussion. • Contact Theatre (2005-2008) for six turntables. References to Books on Applied Mechanics (2006-8) for solo percussionist, electronics and website. Music in the Shape of ELEVEN (2006) for flute, keyboard, string trio, harp, percussion and laptop. Mixtape Zen (2007-2008) for four turntables and percussion. Transference (51.16 North, 1.04 East) (2007-2008), an interactive sound installation for four website projections, microphone and loudspeakers. The works are presented as scores and a DVD-ROM and the supporting commentary, of approximately 20,000 words, deals with the critical context within which these works are set. Drawing upon theories of minimal ism, improvisation, DJ culture, digital media and sonic art, I aim to explain the ways in which each of the works take a unique approach to the concept of 'navigating time.

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

DigitalBeing: an Ambient Intelligent Dance Space.

DigitalBeing is an ambient intelligent system that aims to use stage lighting and lighting in projected imagery within a dance performance to portray dancer’s arousal state. The dance space will be augmented with pressure sensors to track dancers’ movements; dancers will also wear physiological sensors. Sensor data will be passed to a three layered architecture. Layer 1 is composed of a system that analyzes sensor data. Layer 2 is composed of two intelligent lighting systems that use the analyzed sensor information to adapt onstage and virtual lighting to show dancer’s arousal level. Layer 3 translates lighting changes to appropriate lighting board commands as well as rendering commands to render the projected imagery

5-Choosability of Planar-plus-two-edge Graphs

We prove that graphs that can be made planar by deleting two edges are 5-choosable. To arrive at this, first we prove an extension of a theorem of Thomassen. Second, we prove an extension of a theorem Postle and Thomas. The difference between our extensions and the theorems of Thomassen and of Postle and Thomas is that we allow the graph to contain an inner 4-list vertex. We also use a colouring technique from two papers by Dvořák, Lidický and Škrekovski, and independently by Compos and Havet