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# Packings and realizations of degree sequences with specified substructures

#### 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.

https://digitalcommons.unl.edu/dissertations/AAI3450118