Auburn University Graduate School

Partition-good, Recursively Partitionable, and, In Between, Arbitrarily Partitionable

2026-07-24T00:00:00Z

Partition-good, Recursively Partitionable, and, In Between, Arbitrarily Partitionable Nochumson, Shayne Graph partitioning problems form a broad class of problems in graph theory that investigate the partitioning of a graph's vertices, edges, or both into subgraphs satisfying prescribed constraints. This dissertation has three sections, two of which concern graph partitioning and a third that emerged naturally from the investigation of the first two. After introducing the necessary background and historical context, the dissertation further develops the theory of partition-good graphs alongside the well established theory of recursively and arbitrarily partitionable graphs. New measures of partitionability, namely the 2p-number and the web number, are introduced and investigated. The first question of this dissertation works on establishing conditions under which graphs from various classes satisfy partition-goodness and, where appropriate, characterizes conditions for recursive and arbitrary partitionability. Additional results concerning 2p-numbers and web numbers for selected graph classes are also presented when discussing this question. The second question question studied examines under what conditions two disjoint graphs, one traceable and the other connected, may be joined by a single edge such that the resultant graph is partition-good. Finally, motivated by the investigations of the first two sections, the dissertation addresses the enumeration of isomorphism classes of spanning trees of complete bipartite graphs.

Orientations with forbidden out-degree and total-coloring extensions

2026-07-24T00:00:00Z

Orientations with forbidden out-degree and total-coloring extensions Henderschedt, Owen This thesis studies two topics in graph theory: graph orientations with forbidden out-degrees and extensions of total-colorings. An orientation of a graph assigns a direction to each edge, thereby prescribing for each vertex the number of edges directed away from it, called its out-degree. A natural question is to determine when it is possible to orient a graph so that each vertex avoids a prescribed set of forbidden out-degrees. More formally, given a graph $G$ and a function $F:V(G)\to 2^{\mathbb{N}}$, an $F$-avoiding orientation is one in which no vertex $v$ has out-degree in its forbidden set $F(v)$. Akbari et al.\ conjectured that when $|F(v)|< \frac{1}{2}\deg_G(v)$ for all $v$, then $G$ admits an $F$-avoiding orientation. This conjecture remains open even when $G$ is regular and all vertices share the same forbidden list. We extend known results on this problem by verifying the conjecture when $G$ is $5$- and $6$-regular, as well as confirming the general conjecture in several additional cases, including for sparse graph classes and structured families of forbidden sets. The second part of the thesis focuses on coloring extension problems, which ask whether a given partial coloring can be extended to a coloring of the entire graph without introducing new colors. A total-coloring of a graph assigns colors to both its vertices and edges so that no adjacent or incident elements receive the same color. A central open problem in this area, the Total Coloring Conjecture, asserts that every graph admits a total-coloring using at most $\Delta+2$ colors, where $\Delta$ is the maximum degree. We initiate a systematic study of total-coloring extensions by formulating new conjectures inspired by the literature on vertex- and edge-coloring extensions. Our conjectures strengthen the Total-Coloring Conjecture, and thus natural candidate graph classes for their study are those for which the Total-Coloring Conjecture is known to hold. To this end, we establish extension results for planar graphs with large maximum degree. We also prove total-coloring extension results for several other families of graphs.

Transport-Based Generative Modeling for Stochastic Dynamics, Uncertainty Quantification, and High-Dimensional Sampling

2026-07-24T00:00:00Z

Transport-Based Generative Modeling for Stochastic Dynamics, Uncertainty Quantification, and High-Dimensional Sampling Wang, Pengjun This dissertation develops and analyzes generative modeling methods for scientific computing, where repeated sampling from complex probability distributions is needed for uncertainty quantification, estimation of quantities of interest, and statistical inference. The main contributions are pseudo-reversible normalizing flows and training-free diffusion models, which provide two complementary transport-based approaches for generating samples from target distributions by transforming samples from simple reference distributions. In the first part of the dissertation, we present a pseudo-reversible normalizing flow method for efficiently generating samples of the state of a stochastic differential equation (SDE) with various initial distributions. The main novelty of our normalizing flow model is that it can learn the conditional distribution of the state, i.e., the distribution of the final state conditional on any initial state, such that the model only needs to be trained once and the trained model can be used to handle various initial distributions. We provide a rigorous convergence analysis of the learned distribution to the target distribution in the Kullback–Leibler divergence metric. Numerical experiments are provided to demonstrate the effectiveness of the proposed normalizing flow model. Next, we extend this framework to uncertainty quantification for noisy physical models. A conditional pseudo-reversible normalizing flow is constructed directly from input-output data to generate samples from conditional distributions relevant to forward and inverse uncertainty propagation. The method does not require prior knowledge of the deterministic model component or the distribution of additive noise. Theoretical results establish convergence for the forward conditional distribution, and numerical experiments include benchmark examples, high-dimensional uncertainty propagation, and a geologic carbon storage application. In this dissertation, we also study a training-free diffusion model for high-dimensional sampling. A Gaussian mixture model constructed from available samples yields an analytically tractable score function and avoids propagating score function approximation errors through the reverse process. The resulting error bounds exhibit favorable dimension dependence, scaling as $\mathcal{O}(d)$ in the $\ell_2$ norm and $\mathcal{O}(\log d)$ in the $\ell_\infty$ norm. Importantly, the proposed error estimates are fully numerically verifiable with respect to both time-step size and dimensionality, thereby bridging the gap between theoretical analysis and observed numerical behavior.

Food web structure in the Alabama River: Stable isotope-based characterization of a large, dam-regulated southeastern U.S. coastal plain river system

2026-07-24T00:00:00Z

Food web structure in the Alabama River: Stable isotope-based characterization of a large, dam-regulated southeastern U.S. coastal plain river system Norrid, Kate Riverine food webs are influenced by dams, which modify flow regimes, sediment transport, and nutrient cycling, reducing longitudinal connectivity and potentially altering trophic interactions. This study used carbon (δ13C) and nitrogen (δ15N) stable isotopes to characterize trophic relationships within the hydrologically altered Alabama River and evaluate how three lock-and-dam structures influence food web dynamics. Specific objectives were to (1) characterize isotopic patterns in pelagic and benthic resources across sites, (2) compare inter- and intra-specific variation in trophic position among five fish species, and (3) compare inter- and intra-specific variation in trophic (isotopic) niche among the same five species. A total of 2,795 samples, including five fish species, sediment, seston, and macroinvertebrates, were collected from six sites immediately upstream and downstream of the three lock-and-dam structures. Consumer δ13C values showed extensive overlap among taxa, suggesting shared reliance on similar basal carbon resources. In contrast, isotopic niche size was larger downstream than upstream for nearly every species-dam comparison, suggesting that intraspecific niche dynamics captured ecological responses to river regulation not apparent in overall trophic structure. Baseline δ15N values varied longitudinally, exhibiting an overall downstream decline across the river continuum. Despite this longitudinal shift in baseline δ15N, trophic position estimates showed limited variation among adjacent dam-paired sites. These results suggest that the Alabama River supports a broadly integrated food web despite hydrologic alteration and demonstrate the value of intraspecific isotopic niche variation for understanding ecological responses to river regulation.