Using Euler’s Formula to Find the Lower Bound of the Page Number

Abstract

The concept of book embedding, originating in computer science, has found extensive applications in various problem domains. A book embedding of a graph G involves arranging the vertices of G in an order along a line and assigning the edges to one or more half-planes. The page number of a graph is the smallest possible number of half-planes for any book embedding of the graph. Determining the page number is a key aspect of book embedding and carries significant importance. This paper aims to investigate the non-trivial lower bound of the page number for both a graph G and a random graph $G\in \mathcal {G}(n,p)$ by incorporating two seemingly unrelated concepts: edge-arboricity and Euler’s Formula. Our analysis reveals that for a graph G, which is not a path, $pn(G)\ge \lceil \frac{1}{3} a_1(G)\rceil$, where $a_1(G)$ denotes the edge-arboricity of G, and for an outerplanar graph, the lower bound is optimal. For $G\in \mathcal {G}(n,p)$, $pn(G)\ge \lceil \frac{1}{6}np(1-o(1))\rceil$ with high probability, as long as $\frac{c}{n}\le p\le \frac{\root 2 \of {3(n-1)}}{n\log {n}}$.