Off-campus UNL users: To download campus access dissertations, please use the following link to log into our proxy server with your NU ID and password. When you are done browsing please remember to return to this page and log out.

Non-UNL users: Please talk to your librarian about requesting this dissertation through interlibrary loan.

Packings and realizations of degree sequences with specified substructures

Tyler Seacrest, University of Nebraska - Lincoln

Abstract

This thesis focuses on the intersection of two classical and fundamental areas in graph theory: graph packing and degree sequences. The question of packing degree sequences lies naturally in this intersection, asking when degree sequences have edge-disjoint realizations on the same vertex set. The most significant result in this area is Kundu’s k-Factor Theorem, which characterizes when a degree sequence packs with a constant sequence. We prove a series of results in this spirit, and we particularly search for realizations of degree sequences with edge-disjoint 1-factors.^ Perhaps the most fundamental result in degree sequence theory is the Erdös-Gallai Theorem, characterizing when a degree sequence has a realization. After exploring degree sequence packing, we develop several proofs of this famous theorem, connecting it to many other important graph theory concepts. ^ We are also interested in locating edge-disjoint 1-factors in dense graphs. Before tackling this question, we build on the work of Katerinis to find the largest k such that a graph has a k-factor, where the value of k depends on the minimum degree of the graph. This gives an upper bound on the number of edge-disjoint 1-factors. ^ The question of finding edge-disjoint 1-factors leads us to a conjecture of Bollobás and Scott about finding spanning balanced bipartite subgraphs with vertices of high degree. We first prove a degree-sequence version of the Bollobás–Scott Conjecture which we apply to the question of edge-disjoint 1-factors. We then generalize and prove an approximate version of the conjecture, yielding balanced partitions with many edges going to each part. This version has many applications, including finding edge-disjoint 1-factors and edge-disjoint Hamiltonian cycles.^

Subject Area

Theoretical Mathematics

Recommended Citation

Seacrest, Tyler, "Packings and realizations of degree sequences with specified substructures" (2011). ETD collection for University of Nebraska - Lincoln. AAI3450118.
http://digitalcommons.unl.edu/dissertations/AAI3450118

Share

COinS