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- Faculty Publications & Research (4)
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- Hua Kun Liu (1)
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Articles 1 - 23 of 23
Full-Text Articles in Physical Sciences and Mathematics
The Fundamental Groupoid In Discrete Homotopy Theory, Udit Ajit Mavinkurve
The Fundamental Groupoid In Discrete Homotopy Theory, Udit Ajit Mavinkurve
Electronic Thesis and Dissertation Repository
Discrete homotopy theory is a homotopy theory designed for studying graphs and for detecting combinatorial, rather than topological, “holes”. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis.
In this thesis, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us …
On The Singular Pebbling Number Of A Graph, Harmony R. Morris
On The Singular Pebbling Number Of A Graph, Harmony R. Morris
Rose-Hulman Undergraduate Mathematics Journal
In this paper, we define a new parameter of a connected graph as a spin-off of the pebbling number (which is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble). This new parameter is the singular pebbling number, the smallest t such that a player can be given any configuration of at least t pebbles and any target vertex and can successfully move pebbles so that exactly one pebble ends on the target vertex. We also prove that the singular pebbling number of any graph on 3 or more vertices is equal …
Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson
Surjectivity Of The Wahl Map On Cubic Graphs, Angela C. Hanson
Theses and Dissertations--Mathematics
Much of algebraic geometry is the study of curves. One tool we use to study curves is whether they can be embedded in a K3 surface or not. If the Wahl map is surjective on a curve, that curve cannot be embedded in a K3 surface. Therefore, studying if the Wahl map is surjective for a particular curve gives us more insight into the properties of that curve. We simplify this problem by converting graph curves to dual graphs. Then the information for graphs can be used to study the underlying curves. We will discuss conditions for the Wahl map …
Characterizations Of Certain Classes Of Graphs And Matroids, Jagdeep Singh
Characterizations Of Certain Classes Of Graphs And Matroids, Jagdeep Singh
LSU Doctoral Dissertations
``If a theorem about graphs can be expressed in terms of edges and cycles only, it probably exemplifies a more general theorem about matroids." Most of my work draws inspiration from this assertion, made by Tutte in 1979.
In 2004, Ehrenfeucht, Harju and Rozenberg proved that all graphs can be constructed from complete graphs via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. In Chapter 2, we consider the binary matroid analogue of each of these graph operations. We prove that the analogue of the result of Ehrenfeucht et. al. does …
Polychromatic Colorings Of Certain Subgraphs Of Complete Graphs And Maximum Densities Of Substructures Of A Hypercube, Ryan Tyler Hansen
Polychromatic Colorings Of Certain Subgraphs Of Complete Graphs And Maximum Densities Of Substructures Of A Hypercube, Ryan Tyler Hansen
Graduate Theses, Dissertations, and Problem Reports
If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, polyHG, is the largest number of colors in an H-polychromatic coloring. We determine polyHG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q. …
Categorical Aspects Of Graphs, Jacob D. Ender
Categorical Aspects Of Graphs, Jacob D. Ender
Undergraduate Student Research Internships Conference
In this article, we introduce a categorical characterization of directed and undirected graphs, and explore subcategories of reflexive and simple graphs. We show that there are a number of adjunctions between such subcategories, exploring varying combinations of graph types.
On Hamilton Cycle Decompositions Of Complete Multipartite Graphs Which Are Both Cyclic And Symmetric, Fatima A. Akinola
On Hamilton Cycle Decompositions Of Complete Multipartite Graphs Which Are Both Cyclic And Symmetric, Fatima A. Akinola
Theses, Dissertations and Capstones
Let G be a graph with v vertices. A Hamilton cycle of a graph is a collection of edges which create a cycle using every vertex. A Hamilton cycle decomposition is cyclic if the set of cycle is invariant under a full length permutation of the vertex set. We say a decomposition is symmetric if all the cycles are invariant under an appropriate power of the full length permutation. Such decompositions are known to exist for complete graphs and families of other graphs. In this work, we show the existence of cyclic n-symmetric Hamilton cycle decompositions of a family …
Graph-Theoretic Simplicial Complexes, Hajos-Type Constructions, And K-Matchings, Julianne Vega
Graph-Theoretic Simplicial Complexes, Hajos-Type Constructions, And K-Matchings, Julianne Vega
Theses and Dissertations--Mathematics
A graph property is monotone if it is closed under the removal of edges and vertices. Given a graph G and a monotone graph property P, one can associate to the pair (G,P) a simplicial complex, which serves as a way to encode graph properties within faces of a topological space. We study these graph-theoretic simplicial complexes using combinatorial and topological approaches as a way to inform our understanding of the graphs and their properties.
In this dissertation, we study two families of simplicial complexes: (1) neighborhood complexes and (2) k-matching complexes. A neighborhood complex is a simplicial …
The Graphs And Matroids Whose Only Odd Circuits Are Small, Kristen Nicole Wetzler
The Graphs And Matroids Whose Only Odd Circuits Are Small, Kristen Nicole Wetzler
LSU Doctoral Dissertations
This thesis is motivated by a graph-theoretical result of Maffray, which states that a 2-connected graph with no odd cycles exceeding length 3 is bipartite, is isomorphic to K_4, or is a collection of triangles glued together along a common edge. We first prove that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M is M(K_4) or F_7, or M is the cycle matroid of a graph consisting of a collection of triangles glued together along a common edge. This result implies that a 2-connected loopless graph G …
Colorings Of Hamming-Distance Graphs, Isaiah H. Harney
Colorings Of Hamming-Distance Graphs, Isaiah H. Harney
Theses and Dissertations--Mathematics
Hamming-distance graphs arise naturally in the study of error-correcting codes and have been utilized by several authors to provide new proofs for (and in some cases improve) known bounds on the size of block codes. We study various standard graph properties of the Hamming-distance graphs with special emphasis placed on the chromatic number. A notion of robustness is defined for colorings of these graphs based on the tolerance of swapping colors along an edge without destroying the properness of the coloring, and a complete characterization of the maximally robust colorings is given for certain parameters. Additionally, explorations are made into …
Planar Graphs, Biplanar Graphs And Graph Thickness, Sean M. Hearon
Planar Graphs, Biplanar Graphs And Graph Thickness, Sean M. Hearon
Electronic Theses, Projects, and Dissertations
A graph is planar if it can be drawn on a piece of paper such that no two edges cross. The smallest complete and complete bipartite graphs that are not planar are K5 and K{3,3}. A biplanar graph is a graph whose edges can be colored using red and blue such that the red edges induce a planar subgraph and the blue edges induce a planar subgraph. In this thesis, we determine the smallest complete and complete bipartite graphs that are not biplanar.
Graphs Of Classroom Networks, Rebecca Holliday
Graphs Of Classroom Networks, Rebecca Holliday
Electronic Theses and Dissertations
In this work, we use the Havel-Hakimi algorithm to visualize data collected from students to investigate classroom networks. The Havel-Hakimi algorithm uses a recursive method to create a simple graph from a graphical degree sequence. In this case, the degree sequence is a representation of the students in a classroom, and we use the number of peers with whom a student studied or collaborated to determine the degree of each. We expand upon the Havel-Hakimi algorithm by coding a program in MATLAB that generates random graphs with the same degree sequence. Then, we run another algorithm to find the isomorphism …
Cyclic Matching Sequencibility Of Graphs, Richard A. Brualdi, Kathleen P. Kiernan, Seth A. Meyer, Michael W. Schroeder
Cyclic Matching Sequencibility Of Graphs, Richard A. Brualdi, Kathleen P. Kiernan, Seth A. Meyer, Michael W. Schroeder
Mathematics Faculty Research
We define the cyclic matching sequencibility of a graph to be the largest integer d such that there exists a cyclic ordering of its edges so that every d consecutive edges in the cyclic ordering form a matching. We show that the cyclic matching sequencibility of K2m and K2m+1 equals m − 1.
Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West
Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West
Noah Prince
A Roman dominating function of a graph G is a labeling f: V(G) →{0,1,2} such that every vertex with a label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ʋϵV(G)f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(G) + …
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Hua Kun Liu
Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …
Omnisculptures., Cihan Eroglu
Omnisculptures., Cihan Eroglu
Electronic Theses and Dissertations
In this thesis we will study conditions for the existence of minimal sized omnipatterns in higher dimensions. We will introduce recent work conducted on one dimensional and two dimensional patterns known as omnisequences and omnimosaics, respectively. These have been studied by Abraham et al [3] and Banks et al [2]. The three dimensional patterns we study are called omnisculptures, and will be the focus of this thesis. A (K,a) omnisequence of length n is a string of letters that contains each of the ak words of length k over [A]={1,2,...a} as a substring. …
The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz
The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz
Faculty Publications & Research
Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S,T) of the vertex set V(G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t . Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.
Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West
Extremal Problems For Roman Domination, E. W. Chambers, W. Kinnersley, N. Prince, D. B. West
Faculty Publications & Research
A Roman dominating function of a graph G is a labeling f: V(G) →{0,1,2} such that every vertex with a label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ʋϵV(G)f(v) over such functions. Let G be a connected n-vertex graph. We prove that γR(G) ≤ 4n/5, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for γR(G) + …
The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz
The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz
Noah Prince
Erdös and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S,T) of the vertex set V(G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t . Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.
On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince
On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince
Faculty Publications & Research
Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for 'balanced' H random graphs provide extremal examples and determine the extremal function. Examples of 'unbalanced graphs' are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) and made a conjecture on the order of magnitude of D(Ks,t ) for a fixed s and …
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Faculty Publications & Research
Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Highly Connected Multicoloured Subgraphs Of Multicoloured Graphs, H. Liu, R. Morris, N. Prince
Noah Prince
Suppose the edges of the complete graph on n vertices, E(Kn), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s is greater than or equal to 2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4 and that …
On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince
On KS,T-Minors In Graphs With Given Average Degree, A. V. Kostochka, N. Prince
Noah Prince
Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for 'balanced' H random graphs provide extremal examples and determine the extremal function. Examples of 'unbalanced graphs' are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) and made a conjecture on the order of magnitude of D(Ks,t ) for a fixed s and …