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

Abstract

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. ARTICLE INFO: Received : 13 July 2020 Received after revision : 29 August 2020 Available online : 5 January 2021