On the page number of complete odd-partite graphs

A book embedding of a graph G is a pair consisting of an edge-coloring of G and a circular ordering of the vertices of G , such that if we place the vertices in the given ordering on a circle and draw the edges as straight lines in the inner part of the circle, then no two edges of the same color cross. The page number p ( G ) is the smallest number of colors within which G can be book embedded. We generalize the matrix representation of Muder, Weaver, and West for book embeddings of complete bipartite graphs to a representation for all graphs. Furthermore, we prove a new upper bound for the page number of complete odd-partite graphs.