BOOK EMBEDDING OF GRAPHS ON THE PROJECTIVE PLANE

For a positive integer k, a book (with k pages) is a topological space consisting of a spine, which is a line, and k pages, which are half-planes with the spine as their boundary. We say that a graph G admits a k-page book embedding or is k-page book embeddable if there exists a linear ordering of the vertices on the spine and one can assign the edges of G to k pages such that no two edges of the same page cross. Yannakakis proved that every plane graph admits a 4-page book embedding. In this paper, we improve this to graphs on the projective plane, that is, those embedded on the projective plane without edge-crossings. Nakamoto and Nozawa showed that every graph on the projective plane admits a 9-page book embedding. In this paper, we improve the latter result to 6-page embedding. Furthermore, we also prove that every graph on the projective plane admits a 3-page book embedding if it is 5-connected and a 5-page book embedding if it is 4-connected. Our idea of the proofs is to use a Tutte path, which is different from previous ones.