Graduate STEM Fellow Profile

Timothy R. Morris

Project Title: Transforming Experiences
Thesis: Forbidden Subgraphs That Imply Certain Hamiltonian Properties
College/University: University of Colorado Denver
Research Advisor: Michael Ferrara
Degree Sought: Ph.D. Applied Discrete Mathematics
Department: Department of Mathematical and Statistical Sciences
Research Focus: Structures within graphs and what conditions will guarantee that those structures exist.
Teaching Partner(s):

Description of Research

Given a family of graphs F, a graph G is said to be F-free if G does not contain any graph from F as an induced subgraph. G is hamiltonian if G has a cycle of length the order of G. G is pancyclic if G contains cycles of all lengths from 3 to the order of G. Thus the concept of pancyclicity is an extension of the concept of hamiltonicity. The concept of hamiltonicity has been extensively studied in terms of degree conditions, the relationship between connectivity and independence, and forbidden subgraphs. My research extends the forbidden subgraph condition to determine similar conditions that guarantee pancyclicity. For example, I have shown that graphs which are 4-connected, claw-free, and P10-free are pancyclic.

Example of how my research is integrated into my GK-12 experience

One example integrating my research into the classroom was a discussion of graph decompositions. I was able to define for the students a few types of graphs and discussed how to determine the minimum number of graphs of a particular type necessary to decompose another type of graph. This concept was then tied into molecular models and we discussed how this type of information could be used to determine physical properties of such molecules. This discussion showed students that mathematical modeling is useful for scientific understanding.