This is an updated version of Y. P. Liu’s *Topological Theory on Graphs* published by the University of Science and Technology of China Press in 2008. Like the first edition, this book mainly reflects developments established by the author.

The book does not begin well, explaining the meaning of familiar symbols such as \(\vee\), \(\Rightarrow\), \(\exists\), as well as that the notation \((i.j.k)\) will refer to item *k* of section *j* in Chapter *i*. It then, still on page 1, presents some basic laws of set theory, but this all takes place within some undefined set \(\Omega\), so for example the complement of a set is simply defined to be the complement in \(\Omega\). This lack of clarity does much to undermine the reader’s faith in the extreme formality that is to come in the rest of the book.

Important theorems have a way of popping up with no context or proof or historical reference, such as Kuratowski’s famous theorem on planar graphs that appears as Theorem 1.3.9: A graph is planar if and only if it has no subgraph homeomorphic to \(K_5\) or \(K_{3,3}\). (Kuratowski is briefly mentioned in a note 151 pages later, not that his theorem is mentioned there.) Two earlier theorems that also appear with no proof or discussion are the characterization of Eulerian graphs having only even vertices, and bipartite graphs having no odd cicuits.

The book claims to solve a number of problems such as the embeddability of a graph on a surface with given genus, the Gauss crossing conjecture, and the graphicness and cographicness of a matroid. Several results follow: theorems of Lefschetz on double coverings, MacLane on cycle bases, and Whitney on duality for planarity.

John J. Watkins is Professor Emeritus of Mathematics at Colorado College.