Abstract:
Start with a graph whose edges are labeled by positive integers. Label each vertex with an integer so that if two vertices are joined by an edge the vertex labels are congruent modulo the edge label. A set of vertex labels satisfying this condition is called a spline. Much of our research pertains to splines on n-cycles. We use the Chinese Remainder Theorem to find particularly nice splines, and we identify two bases for the set of splines on n-cycles.