Periodic Steiner Networks Minimizing Length

Abstract

We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices K_4. For comparison we consider a competing topological class, also with a quotient on four vertices, and determine the minimizing ths networks.