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Learn Graph Theory

Dots and lines that model networks, maps, schedules and the internet.

Free to start. Built from a single prompt. No credit card.

Graph theory studies networks made of vertices joined by edges. That simple model describes road maps, friendships, computer networks, molecules and course prerequisites. The subject began with Euler's 1736 solution to the Konigsberg bridges puzzle, and its core ideas are surprisingly accessible: degrees, paths and cycles, trees, coloring and planar drawings. It is also the mathematics behind algorithms that find shortest routes and schedule tasks. Many results come with short, elegant proofs, so it is a friendly place to start proof-based math. Learnr builds a graph theory course for your level and teaches it through short interactive lessons full of small graphs to draw.

How to learn Graph Theory, step by step

Draw small graphs constantly. Sketch every example, from the complete graph K4 to a five-vertex cycle. Counting degrees on paper shows why the sum of all degrees always equals twice the number of edges.
Solve the Konigsberg bridges problem yourself. Model the land masses as vertices and bridges as edges, then prove that a connected graph has an Euler circuit exactly when every vertex has even degree. It is a perfect first proof.
Learn trees until they feel obvious. Prove that a tree with n vertices has exactly n - 1 edges and a unique path between any two vertices. Trees appear everywhere later, from file systems to spanning networks.
Practice coloring and bipartite graphs. Show a graph is bipartite exactly when it has no odd cycle, then find chromatic numbers of small graphs. Map coloring and exam scheduling make good real problems for this.
Run the classic algorithms by hand. Trace Dijkstra's shortest-path algorithm and Kruskal's minimum spanning tree on a weighted graph of about eight vertices, keeping a table of every step so you see why each choice is safe.
Review with short daily problem sets. Graph theory rewards seeing many examples. A few problems a day in a Learnr course, kept going by a streak, builds the pattern recognition that makes new proofs feel approachable.

What you'll learn

Use vertices, edges, degrees, paths and cycles precisely
Prove the handshake lemma and basic facts about trees
Decide when a graph has an Euler circuit or path
Color graphs and reason about planarity
Run shortest-path and spanning-tree algorithms by hand

What's inside your Graph Theory course

A sample path Learnr builds for Graph Theory. Your real course adapts to your goal and level.

1. Graphs, degrees and the handshake lemma
2. Walks, paths, cycles and connectivity
3. Trees and spanning trees
4. Euler and Hamilton circuits
5. Coloring, matchings and bipartite graphs
6. Planar graphs and graph algorithms

How long does it take to learn Graph Theory?

The core of introductory graph theory takes about 6 to 12 weeks at 4 to 6 hours a week. A full university course with proofs on matchings, connectivity and planarity runs about a semester, and less if you already write proofs.

What you need before you start

Basic algebra and a willingness to learn simple proofs, especially induction. Discrete math helps, but many introductions to graph theory teach the needed proof techniques along the way.

Common mistakes to avoid

Mixing up Euler circuits, which use every edge once, with Hamilton cycles, which visit every vertex once. Only the Euler version has a simple degree test.
Assuming a drawing with crossing edges means a graph is not planar. Try redrawing it before concluding anything.
Proving claims from one or two drawn examples. Test against small counterexamples, then write a general argument, often by induction on vertices.

How Learnr builds your course

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Graph Theory questions, answered

Is graph theory the same as graphing functions?

No. Graph theory is about networks of vertices and edges, not plotting y against x on axes. The shared word is a historical coincidence.

Is graph theory useful for programming?

Very. Social networks, routing, dependency resolution and recommendation systems all rely on graph algorithms, and graph questions are common in technical interviews.

What is the four color theorem?

It states that any map drawn on a flat plane can be colored with at most four colors so neighboring regions differ. It was proved in 1976 with substantial computer assistance.