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On dispersability of some circulant graphs

Paul C. Kainen, Samuel S. Joslin, Shannon Overbay

Abstract

The matching book thickness of a graph is the least number of pages in a book embedding such that each page is a matching. A graph is dispersable if its matching book thickness equals its maximum degree. Minimum page matching book embeddings are given for bipartite and for most non-bipartite circulants contained in the (Harary) cube of a cycle and for various higher-powers.

On dispersability of some circulant graphs

Abstract

The matching book thickness of a graph is the least number of pages in a book embedding such that each page is a matching. A graph is dispersable if its matching book thickness equals its maximum degree. Minimum page matching book embeddings are given for bipartite and for most non-bipartite circulants contained in the (Harary) cube of a cycle and for various higher-powers.

Paper Structure

This paper contains 8 sections, 11 theorems, 14 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. The bipartite case of $C(n,\{1,3\})$
  4. The nonbipartite case of $C(n, \{1,3\})$
  5. $C(n,\{1,2\})$ is nearly dispersable
  6. The case $C(n,\{2,3\})$
  7. Larger degree and jump-lengths
  8. Discussion

Key Result

Lemma 1

Let $G$ be a regular nearly dispersable graph of order $n$. Then the sparseness of $G$ is at least 1 if $n$ is even and at least $\Delta/2$ if $n$ is odd.

Figures (14)

  • Figure 1: Common four-coloring $c$ of the twist
  • Figure 2: Dispersable embeddings of $C(8,\{1,3\})$ and $C(10,\{1,3\})$
  • Figure 3: Dispersable embeddings of $C(16,\{1,3\})$ and $C(18,\{1,3\})$
  • Figure 4: Nearly dispersable embeddings of $C(13,\{1,3\})$ and $C(11,\{1,3\})$
  • Figure 5: $C(7,\{1,2\})$, $C(8,\{1,2\})$$C(10,\{1,2\})$ are nearly dispersable.
  • ...and 9 more figures

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • ...and 12 more