Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

A principal tool used in construction of crossing-critical graphs are tiles. In the tile concept, tiles can be arranged by gluing one tile to another in a linear or circular fashion. The series of tiles with circular fashion form an infinite graph family. In this way, the intersection number of this family of graphs can be determined. In this research, has been formed an infinite family graphs Q_((1,s,b) ) (n) with average degree r between 3.5 and 4. The graph formed by gluing together many copies of the tile P_((1,s,b) ) in circular fashion, where the tile P_((1,s,b) ) consist of two identical pieces of tile. And then, the graph embedded into the book to determine the pagenumber that can be formed. When embed graph into book, the vertices are put on a line called the spine and the edges are put on half-planes called the pages. The results obtained show that the graph Q_((1,s,b) ) (n) has 10-crossing-critical and book embedding of graph has 4-page book.

On Open Book Embedding of Contact Manifolds in the Standard Contact Sphere

We prove some open book embedding results in the contact category with a constructive approach. As a consequence, we give an alternative proof of a theorem of Etnyre and Lekili that produces a large class of contact 3-manifolds admitting contact open book embeddings in the standard contact 5-sphere. We also show that all the Ustilovsky $(4m+1)$-spheres contact open book embed in the standard contact $(4m+3)$-sphere.