Intersection Graphs of Translated Regular Polygons

Intersection graphs are graphs in which each vertex represents a set and two vertices are adjacent if and only if the sets intersect. Our research focuses on intersection graphs in which the sets in question are translated regular polygons in the plane. There a variety of methods for answering questions about these graphs, which draw from both geometry and graph theory. We prove that for odd $n$, the graphs representable with regular $n$-gons are exactly those representable with regular $2n$-gons. We also show that there are graphs representable with triangles (and hexagons) but not squares, and discuss our investigation as to whether there are graphs representable with squares but not triangles.