Module Bases for Integer Splines

Abstract:  Let G be a graph with n vertices and integer edge weights A.  A vertex labeling (g_1,...g_n) is an integer spline if for every pair of adjacent vertices {i,j} with edge weight a, g_i = g_j mod a.  It turns out that S(G,A) forms a free module over the integers of rank n, which means there exists a basis with n elements.  We provide an explicit construction for a particularly nice basis for this module, generalizing work done by  undergraduates at Smith College and Bard College.