Polytopes and graphs
This page is dedicated to my book Polytopes and Graphs. This material has been published by Cambridge University Press as ‘Polytopes and Graphs’ by Guillermo Pineda Villavicencio. A pre-publication copy of the book is available for personal use only from Deakin Research Online. This version is free to view and download for personal use only. It is not for redistribution, resale, or use in derivative works. © Guillermo Pineda Villavicencio 2018-2024.
‘Polytopes and Graphs’ places graphs at the heart of the study of polytopes. It covers topics on the graphs of polytopes and explores how these graphs elucidate aspects of the combinatorial structure of polytopes. The book seeks to concretely address two key questions:
- Given the graph of a polytope, what properties can we infer about the polytope?
- Given a class of polytopes, what properties characterise their graphs?
Errata to the first version of the book
Pages refer to the official ebook
CHAPTER 2
1. Page 58: ‘The undirected graph formed by the vertices and edges of $P$, denoted by $G(\mathcal C)$,…’ should be ‘… denoted by $G(P)$…’
2. Page 67: The sign function is not in the glossary.
3. Page 105: First sentence of the second paragraph after Corollary 2.10.2, one ‘and’ should be removed.
4. Page 109: In the second-last paragraph, one ‘the complex’ should be removed.
5. Page 120: In the second-last sentence before Section 2.12, ‘propostion’ should be corrected to ‘proposition’.
6. Remark 2.13.2: It should read ‘The Euler–Poincar\’e–Schl\”afli equation (2.12.17)…’ instead of ‘The Euler–Poincar\’e–Schl\”afli equation (Theorem 2.13.3)…’
7. Remark 2.13.4: It should read ‘The Euler–Poincar\’e–Schl\”afli equation (2.12.17)…’ instead of ‘The Euler–Poincar\’e–Schl\”afli equation (Theorem 2.13.3)…’
8. Theorem 2.14.13: Insert ‘(other than the simplex)’ after ‘$d$-polytope’ in the first sentence. Thus, the sentence becomes: ‘Let $\mathcal G$’ be the Gale transform of a $d$-polytope (other than the simplex) on $n$ vertices.
9. Problem 2.15.3: $P=\text{conv}(F\cup F_1)$ should be replaced by $\text{aff} (P)=\text{aff}(F\cup F_1)$.
CHAPTER 3
1. Page 155: The sentence ‘This graph as an abstract is often viewed graph and analysed using methods from graph theory.’ should be ‘This graph is often viewed as an abstract graph and analysed using methods from graph theory.’
2. Page 173: Line 2, $p$ should be replaced by $P$.
3. Page 196: Line 3, vertices $p_{u^{+}}$ to $p_{v^{+}}$ should be in bold, i.e. $\vec p_{u^{+}}$ to $\vec p_{v^{+}}$.
4. Page 218: In Eq (3.9.5.2), ‘:=’ should be replaced by ‘=:’.
5. Page 219: Line 7, ‘:=’ should be replaced by ‘=:’
CHAPTER 5
1. Page 259: Line -2, ‘…is nonvisible from the point’ should be replaced by ‘…is nonvisible from the point $\vec v$’.
CHAPTER 6
1. Page 278: Line -4, the interval ‘[0,\infty]’$ should be replaced by ‘[0,\infty)’$.
2. Page 280: In Fig. 6.2.1(a), the semicolons in vectors ‘\vec v_2:(1,0,-1)^t’ and ‘\vec v_3:(1,0,-1)^t’ should be replaced by equal signs as ‘\vec v_2=(1,0,-1)^t’ and ‘\vec v_3=(1,0,-1)^t’.
Updates on the open problems in the book
As of now, there are no new developments to report.