Stable embeddings on closed surfaces with respect to the minimum length

Abstract

An embedding of a graph on a closed surface with suitable metric is said to be minimum-length embedding if the total sum of lengths of its edges measured by the metric is the minimum among all embeddings isotopic to it and is said to be stable with respect to minimum length if the limit of any convergent sequence of minimum-length embeddings isotopic to it is an embedding of the graph. We shall discuss these notions and shall decide which 4-regular quadrangulations and which 6-regular triangulations on the torus have minimum-length embeddings and are stable with respect to minimum length.