A (2k + 1)-regular graph with page-number k

doi:10.1016/j.disc.2022.113214

Discrete Mathematics

Volume 346, Issue 1, January 2023, 113214

https://doi.org/10.1016/j.disc.2022.113214 Get rights and content

Abstract

Bernhart and Kainen asked whether there is a $(2 k + 1)$ -regular graph with page-number k in the paper Bernhart and Kainen (1979) [1]. We construct the required graphs for $k \geq 2$ .

Introduction

A k-book is a subspace of $R^{3}$ consisting of k half-planes that has a line in common but are pairwise disjoint otherwise, where each half-plane is called a page and the line is said to be the spine. A book embedding of a graph G is an ordering of vertices of G along the spine with a drawing of edges of G on the pages such that edges drawn in the same page do not intersect. The page-number of G, denoted by $p (G)$ , is the smallest k such that G has an embedding in a k-book.

The relationship between the page-number and the degree of a graph is a focus in the study of book embedding. Chung et al. [2] gave upper and lower bounds on the page-number of a graph in which no vertex has degree exceeding some number d. Bernhart and Kainen [1] showed that for $k \geq 1$ , if $p (G) \leq k$ , then $d (G) < 2 k + 2$ , where $d (G)$ is the average degree of the vertices of a graph G. Furthermore, they proved that $χ (G) \leq 2 k + 2$ if $p (G) \leq k$ , where $χ (G)$ is the chromatic number of G. They found that the bound $2 k + 2$ can be improved if for $k \geq 3$ and $p (G) \leq k$ , we could show G had a vertex with degree at most 2k. So they proposed the following problem.

Problem 1.1

[1] Can we find a $(2 k + 1)$ -regular graph G with page-number k?

As a 1-page graph is outerplanar, it contains at least one vertex of degree at most two. So there is not any 3-regular graph with page-number one. In the paper we construct a $(2 k + 1)$ -regular graph with page-number k for $k \geq 2$ .

The arrangement of the paper is as follows. In Section 2 we construct a $(2 k + 1)$ -regular graph with page-number k for $k \geq 2$ . In Section 3 we obtain a 2k-regular graph with page-number k. The rest of the section is contributed for some terminologies and a result of book embedding.

Graphs in the paper are simple. The order of a graph is the number of its vertices. A graph G is d-regular if the degree of each vertex in G is d. A graph is called outerplanar if it has an embedding in the plane such that all vertices are on the boundary of its outer face. A maximal outerplanar graph of order at least three can be embedded in the plane such that the boundary of its outer face is a hamiltonian cycle and the boundary of any other face is a triangle. So a maximal outerplanar graph of order $n (\geq 3)$ has $2 n - 3$ edges.

Suppose that a graph G is embedded in a book. Then the vertices of G have been arranged in the spine in some order. We use ≺ to denote the order in which for any two vertices x and y, $x ≺ y$ if and only if x proceeds y in the order.

The theorem below gives a lower bound on the page-number of a graph.

Theorem 1.2

[1] If G is a graph of order $n (> 3)$ with m edges, then $p (G) \geq ⌈ \frac{m - n}{n - 3} ⌉ .$

Section snippets

A $(2 k + 1)$ -regular graph with page-number k

In the section we first study necessary conditions for a $(2 k + 1)$ -regular graph to have page-number k. The conditions guide our construction of the desired graph.

Theorem 2.1

Let $k \geq 2$ be an integer. If G is a $(2 k + 1)$ -regular graph of order n with page-number k, then $n \geq 6 k$ .

Proof

Obviously, G has $\frac{1}{2} n (2 k + 1)$ edges. By Theorem 1.2, we have that $p (G) \geq ⌈ \frac{\frac{1}{2} n (2 k + 1) - n}{n - 3} ⌉ = ⌈ \frac{2 k n - n}{2 n - 6} ⌉ \geq \frac{2 k n - n}{2 n - 6} .$

Considering that $p (G) = k$ , we have that $k \geq \frac{2 k n - n}{2 n - 6} .$

So $n \geq 6 k$ . □

Theorem 2.2

Let $k \geq 2$ be an integer. If G is a $(2 k + 1)$ -regular graph of order 6k with

Further discussion

We have constructed a $(2 k + 1)$ -regular graph with page-number k for $k \geq 2$ . One might ask whether such a graph is unique for a fixed k. The answer is no. We now give a counterexample for $k = 2$ . In the proof of Lemma 2.4, the icosahedron is a 5-regular graph of order 12 with page-number 2. We now give another 5-regular graph with page-number 2. For $i = 1, 2$ , let $F_{i}$ be the graph shown in (i) in Fig. 13. Let F be the union of $F_{1}$ and $F_{2}$ . It is easy to check that F is a 5-regular graph of order 16 with

Declaration of Competing Interest

The author has not any conflict of interest with any person or any organization for the paper.

Acknowledgements

The author thanks the referees for a careful reading of the manuscript and their helpful suggestions.

References (3)

F. Bernhart et al.
The book thickness of a graph
J. Comb. Theory, Ser. B
(1979)

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A (2k + 1)-regular graph with page-number k