Introduction to Graph Theory by Douglas West. If you're taking a first course in graph theory, this is where you should start. There is more than enough material here for 3 semesters, and, should you finish it all, you will certainly know more than the average grad student. The only disadvantage of this book is that it's getting old. I've asked Doug when there was a new edition coming out and not gotten much of a response, so don't hold your breath. This is a solid intro to the entire field.
Topological graph theory:
Graphs, Groups, and Surfaces by Arthur T. White. Dr. White is one of the leading experts in this subfield. His previous book, Graphs of Groups on Surfaces is also recommended, if a bit pricey. Obscure and OOP mathematical monographs tend to run that way though, so, I suspect if you're interested in this book, that's not much of a problem to you.
This book is the standard textbook in topological graph theory. As I recall, the topology prerequisites are fairly minimal. For algebra, you want some basic familiarity with groups, but I don't recall anything to heavy hitting being used here. One of the main results is an outline of the proof of the Haywood map coloring theorem, which establishes the chromatic number of all orientable and non-orientable surfaces except the sphere/plane.
Introduction to Graph Theory by Douglas West. If you're taking a first course in graph theory, this is where you should start. There is more than enough material here for 3 semesters, and, should you finish it all, you will certainly know more than the average grad student. The only disadvantage of this book is that it's getting old. I've asked Doug when there was a new edition coming out and not gotten much of a response, so don't hold your breath. This is a solid intro to the entire field.
Topological graph theory:
Graphs, Groups, and Surfaces by Arthur T. White. Dr. White is one of the leading experts in this subfield. His previous book, Graphs of Groups on Surfaces is also recommended, if a bit pricey. Obscure and OOP mathematical monographs tend to run that way though, so, I suspect if you're interested in this book, that's not much of a problem to you.
This book is the standard textbook in topological graph theory. As I recall, the topology prerequisites are fairly minimal. For algebra, you want some basic familiarity with groups, but I don't recall anything to heavy hitting being used here. One of the main results is an outline of the proof of the Haywood map coloring theorem, which establishes the chromatic number of all orientable and non-orientable surfaces except the sphere/plane.