--- title: Planar Graphs description: Planar graphs are graphs that can be drawn on a plane without intersecting edges. Learners will understand Euler's formula, Kuratowski's theorem, and how to analyze map colorings and circuit board layouts. category: mathematics subcategory: graph-theory difficulty: beginner, intermediate, advanced url: /subject/planar-graphs --- # Planar Graphs Planar graphs are graphs that can be drawn on a plane without intersecting edges. Learners will understand Euler's formula, Kuratowski's theorem, and how to analyze map colorings and circuit board layouts. ## Available Resources 3 Books • 2 Courses • 2 Websites • 2 Papers ## Websites ### 1. 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. **Difficulty:** Beginner | **Price:** Free **Link:** https://diestel-graph-theory.com/ **Tags:** graph-theory, textbook, discrete-mathematics, combinatorics, free-online ### 2. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Papers ### 1. On the Cutting Edge: Simplified O(n) Planarity by Edge Addition **Author:** John M. Boyer, Wendy J. Myrvold 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. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://jgaa.info/index.php/jgaa/article/view/paper91 **Tags:** planarity-testing, graph-algorithms, planar-embedding, kuratowski-subgraph, linear-time-algorithms ### 2. An Update on the Four-Color Theorem **Author:** Robin Thomas 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://thomas.math.gatech.edu/PAP/update.pdf **Tags:** four-color-theorem, graph-coloring, planar-graphs, discharging-method, reducible-configurations ## Books ### 1. Graphs on Surfaces **Author:** Bojan Mohar, Carsten Thomassen 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0801866898?tag=edmonddante07-20 **Tags:** topological-graph-theory, graph-embedding, graph-genus, kuratowski-theorem, planar-graphs ### 2. Introduction to Graph Theory **Author:** Richard J. Trudeau Dover paperback written for readers with no mathematical background. Five of its eight chapters — planar graphs, Euler's formula, Platonic graphs, coloring, and genus — cover exactly this topic, with complete proofs and end-of-chapter exercises. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0486678709?tag=edmonddante07-20 **Tags:** graph-theory, planar-graphs, euler-formula, graph-coloring, graph-genus ### 3. Graph Theory (Diestel, 6th Edition) — Free Online Edition **Author:** Reinhard Diestel 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. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://diestel-graph-theory.com/basic.html **Tags:** graph-theory, planar-graphs, kuratowski-theorem, plane-duality, graph-minors ## Courses ### 1. 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. **Difficulty:** Beginner | **Language:** English | **Duration:** 6 weeks of study, 6–10 hours per week. | **Price:** Free **Link:** https://www.coursera.org/learn/algorithms-part1 **Tags:** courses, technology-computer-science, algorithms ### 2. Introduction to Graph Theory We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. In this online course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible! By the end of the course, we will implement an algorithm which finds an optimal assignment of students to schools. This algorithm, developed by David Gale and Lloyd S. Shapley, was later recognized by the conferral of Nobel Prize in Economics. As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students. **Difficulty:** Beginner | **Language:** English | **Duration:** 5 weeks, 3-5 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/graphs **Tags:** courses, mathematics-statistics, combinatorics --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/planar-graphs