Mathematics / core
Graphs, connectivity, and structure
Use graphs to reason about paths, cuts, matchings, colorings, planarity, and the algebra of networks.
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Module 01 core
Graphs, connectivity, and structure
Use graphs to reason about paths, cuts, matchings, colorings, planarity, and the algebra of networks.
- 01
Graph models, degrees, and isomorphism
Choose a graph model, double-count its incidences, and distinguish relabeling from a change of structure.
22 min - 02
Walks, paths, and connected components
Turn reachability into an equivalence relation and use breadth-first layers to certify unweighted distances.
23 min - 03
Trees and spanning-tree structure
Relate unique routes, absence of cycles, and minimal connectivity, then isolate a spanning tree inside a larger graph.
23 min - 04
Eulerian circuits and Hamiltonian cycles
Separate visiting every edge from visiting every vertex, prove the Euler parity criterion, and use Hamiltonian obstructions carefully.
24 min - 05
Cuts, connectivity, and Menger's theorem
Measure route redundancy by matching disjoint paths with the smallest separating failure set.
25 min - 06
Bipartite matchings and Hall's theorem
Translate distinct assignments into matchings, detect collective shortages, and improve a matching with alternating paths.
25 min - 07
Flows, cuts, and integral flow
Conserve throughput, improve a flow through residual routes, and certify optimality with a cut of equal capacity.
26 min - 08
Vertex colorings and chromatic polynomials
Encode conflicts with proper colorings, bound the number of needed colors, and count labeled colorings recursively.
25 min - 09
Planar graphs and Euler's formula
Separate a crossing-heavy drawing from nonplanarity, count faces correctly, and derive edge bounds from boundary incidences.
25 min - 10
Graph Laplacians and connectivity
Express edge disagreements as a positive quadratic form and read connected components from the Laplacian kernel.
27 min