A graph is defined as a set of vertices that are connected by edges. Some graphs can be embedded in the plane, i.e. drawn in the plane without any edges crossing. Such graphs are called planar. Two important examples of graphs that are not planar are the complete graph K_{5} and the complete bipartite graph K_{3,3}. See Figures 1 and 2 for an illustration of them. No matter how one moves around their vertices, there will always be at least one pair of edges crossing. What makes K_{5} and K_{3,3} special? Kuratowski's theorem states that a graph is planar if and only if it "contains" neither the graph K_{5} nor the graph K_{3,3}, for a certain definition of "contains". In other words, K_{5} and K_{3,3} are the only two obstacles that can prevent a graph from being planar, and in that sense they belong with each other.

Not being planar also means not being embeddable on the outer surface of a standard (wedding) ring. However, our wedding rings are Moebius rings, which means that they have a 180 degree twist in them. It turns out, that both K_{5} and K_{3,3} can be engraved on such Moebius rings without any edges crossing. And not just that: they can both be engraved in beautifully symmetric ways, yielding our beautiful wedding rings.