Counting Links in Complete Graphs

Abstract

We find the minimal number of non-trivial links in an embedding of any complete $k$-partite graph on 7 vertices (including $K_{7}$, which has at least 21 non-trivial links). We give either exact values or upper and lower bounds for the minimal number of non-trivial links for all complete $k$-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.