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# Combinatorics using computational methods

#### Abstract

Computational combinatorics involves combining pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics. This thesis provides a theoretical framework for combinatorial search, which is then applied to several problems in combinatorics. ^ Chain Counting: Linek asked which numbers can be represented as the number of chains in a width-two poset. By developing a method for counting chains in posets generated from small configurations, constructions are found to represent every number from five to 50 million, providing strong evidence that all numbers are representable. ^ Ramsey Theory on the Integers: Van der Waerden's Theorem states that for sufficiently large *n* the numbers 1, 2, …, * n* cannot be *r*-colored while avoiding monochromatic arithmetic progressions. Finding the minimum *n* with this property is an incredibly difficult problem. We develop methods to compute the minimum *n* as well as optimal colorings when trying to avoid two generalizations of arithmetic progressions. ^ *p*-Extremal Graphs: For an integer *p*, a *p*-extremal graph is a graph with the maximum number of edges over all graphs of order *n* with *p* perfect matchings. We describe the structure of *p*-extremal graphs in terms of a finite number of fundamental graphs and then discover these fundamental graphs using a computational search. ^ Uniquely *K _{r}*-Saturated Graphs: A graph

*G*is uniquely

*K*-saturated if

_{r}*G*contains no copy of

*K*, but adding any missing edge to

_{r}*G*creates exactly one copy of

*K*as a subgraph. Very little was known about uniquely

_{r}*K*-saturated graphs, but by adapting a technique from combinatorial optimization we found several new examples of these graphs. One of these graphs led to the discovery of two new infinite families of uniquely

_{r}*K*-saturated graphs. ^ Some results in space-bounded computational complexity are also presented. First, two nondeterministic complexity classes defined by the number and structure of computation paths are shown to be equal. Second, a log-space algorithm is developed to solve reachability problems on directed graphs that are embedded in surfaces of low genus.^

_{r}#### Subject Area

Mathematics|Computer Science

#### Recommended Citation

Stolee, Derrick, "Combinatorics using computational methods" (2012). *ETD collection for University of Nebraska - Lincoln*. AAI3499340.

https://digitalcommons.unl.edu/dissertations/AAI3499340