Oral Presentations and Performances: Session I
Project Type
Presentation
Project Funding and Affiliations
Mathematical Sciences
Faculty Mentor’s Full Name
Mark Kayll
Faculty Mentor’s Department
Mathematical Sciences
Additional Mentor
Michael Severino, mseverino@fvcc.edu
Abstract / Artist's Statement
The infamous four-color problem (4CP) dates back to 1852 and asks whether four colors suffice to color a map so that countries sharing a border receive distinct colors. This year marks the 50th anniversary of the problem's resolution. The 4CP inspired a bountiful collection of mathematics in the subject now known as "graph theory." A graph models any sort of network such as the Internet, transportation systems, and the neural networks that underpin modern AI. Graphs consist of two components: "vertices" and "edges." We visualize the vertices as dots on a page and the edges as lines connecting pairs of dots. Analogous to the 4CP, coloring the vertices of a graph means to assign each vertex a color so that any two vertices sharing an edge receive distinct colors. Given enough colors, it might be possible to color a graph in numerous ways. In honor of the 4CP, this project seeks to count those ways, taking into account the underlying symmetries. For example, consider a beaded necklace where the beads are vertices and adjacent beads correspond to edges. With colored beads, that necklace may look the same after a rotation or a flip. We want to count the number of distinguishable necklaces. Somewhat surprisingly, a tool from algebra plays a crucial role in determining the desired count.
Category
Physical Sciences
Graph colorings up to symmetries
UC 329
The infamous four-color problem (4CP) dates back to 1852 and asks whether four colors suffice to color a map so that countries sharing a border receive distinct colors. This year marks the 50th anniversary of the problem's resolution. The 4CP inspired a bountiful collection of mathematics in the subject now known as "graph theory." A graph models any sort of network such as the Internet, transportation systems, and the neural networks that underpin modern AI. Graphs consist of two components: "vertices" and "edges." We visualize the vertices as dots on a page and the edges as lines connecting pairs of dots. Analogous to the 4CP, coloring the vertices of a graph means to assign each vertex a color so that any two vertices sharing an edge receive distinct colors. Given enough colors, it might be possible to color a graph in numerous ways. In honor of the 4CP, this project seeks to count those ways, taking into account the underlying symmetries. For example, consider a beaded necklace where the beads are vertices and adjacent beads correspond to edges. With colored beads, that necklace may look the same after a rotation or a flip. We want to count the number of distinguishable necklaces. Somewhat surprisingly, a tool from algebra plays a crucial role in determining the desired count.