Graph Theory (Diestel, 6th Edition) — Free Online Edition
by Reinhard Diestel · Springer (Graduate Texts in Mathematics 173) / diestel-graph-theory.com
Reinhard Diestel's graduate text, readable free online chapter by chapter. Chapter 4 develops planar graphs from the Jordan curve theorem through Euler's formula, Kuratowski's and Wagner's theorems, plane duality, and Tutte's planarity criterion.
More resources on Planar Graphs
Algorithms, Part I
This course covers the essential information that every serious programmer needs to know about algorithms and data structures, with emphasis on applications and scientific performance analysis of Java implementations. Part I covers elementary data structures, sorting, and searching algorithms. Part II focuses on graph- and string-processing algorithms. All the features of this course are available for free. People who are interested in digging deeper into the content may wish to obtain the textbook Algorithms, Fourth Edition (upon which the course is based) or visit the website algs4.cs.princeton.edu for a wealth of additional material. This course does not offer a certificate upon completion.
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Diestel Graph Theory
Official site for Reinhard Diestel's Springer graduate text, where the main text is readable free online and paid eBook editions add the full apparatus. It covers matching, connectivity, planarity, colouring, flows, extremal theory, and minors.
On the Cutting Edge: Simplified O(n) Planarity by Edge Addition
Peer-reviewed paper giving the linear-time planarity test now implemented in Boost Graph Library, NetworkX, SageMath, and JGraphT. Describes the edge-addition method, how it produces either a planar embedding or a Kuratowski subgraph, and why earlier linear algorithms resisted implementation.
An Update on the Four-Color Theorem
Survey by a co-author of the 1997 simplified proof. Traces the history of map coloring, explains why the problem is really about planar graphs, and walks through discharging, reducibility, the 633 configurations, and the resulting quadratic-time coloring algorithm.
Graphs on Surfaces
The reference monograph on topological graph theory. Covers Kuratowski's theorem and other planarity criteria, the Jordan curve theorem and its extensions, rotation systems, graph genus, face-width, and the Robertson–Seymour extension of Kuratowski to arbitrary surfaces. It answers the question every learner asks after Kuratowski — 'what if the surface isn't a plane?' — and is the only book that treats planarity as a special case of embedding rather than as a standalone trick.