#### Date of Award

5-2021

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Patrick Bennett

#### Second Advisor

Dr. Andrzej Dudek

#### Third Advisor

Dr. Alan Frieze

#### Fourth Advisor

Dr. Allen Schwenk

#### Keywords

Graph theory, random structures, random graphs, random algorithms, combinators

#### Abstract

This work addresses two problems in optimizing substructures within larger random structures. In the first, we study the triangle-packing number *v*(*G*), which is the maximum size of a set of edge-disjoint triangles in a graph. In particular we study this parameter for the random graph *G*(*n*,*m*). We analyze a random process called the online triangle packing process in order to bound *v*(*G*). The lower bound on *v*(*G*(*n*,*m*)) that this produces allows for the verification of a conjecture of Tuza for *G*(*n*,*m*). This conjecture states that in any graph *G*, there is a set of edges intersecting every triangle in the graph, and such that the size of this set of edges is bounded above by twice the triangle packing number *v*(*G*). This work is a refinement of the methodology employed by Bennett, Dudek and Zerbib by establishing dynamic concentration of key random variables using the differential equation method. Tuza’s conjecture has been independently verified in the case *G*(*n*, *p*) by Kahn and Park using a very different approach.

In the second problem, we seek to study the number of paths in the *r*-edge-colored random graph *G*(*n*, *p*), where adjacent edges have different colors. This question is inspired by a problem in coding theory and the work of Espig, Frieze, and Krivelevich,who found conditions under which a random graph with randomly 2-colored-edges has a path that alternates between the two colors. Here the parameter alternating connectivity, *kr,l**(G*) is studied for random graphs. The parameter *kr,l* (*G*) is the maximum* t *such that there is an *r*-edge-coloring of *G *such that any pair of vertices is connected by* t *internally disjoint and alternating (i.e. no consecutive edges of the same color) paths of length *l.* We track this parameter’s behavior in *G*(*n*, *p*) over various ranges of *p *by utilizing different strategies and results for each range.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Cushman, Ryan, "On Problems in Random Structures" (2021). *Dissertations*. 3713.

https://scholarworks.wmich.edu/dissertations/3713