Alternating diagrams of 4-regular graphs in 3-space

Embeddings of 4-regular graphs into 3-space are examined by studying graph diagrams, i.e., projections of embedded graphs to an appropriate plane. An invariant of graph diagrams, first introduced by Yokota, is re-formulated and used to show that reduced alternating diagrams have minimal crossing number. The results presented here extend some of the so-called Tait Conjectures.