Graph theory & combinatorics
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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$1,162 · Graph theory & combinatorics · 18 tutoring hrs
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Topics typically covered
Standard syllabus
Graph fundamentals
- Graphs, digraphs, and basic terminology (vertices, edges, degree)
- Paths, cycles, connectivity, and components
- Trees and spanning trees; Cayley's formula (statement)
- Euler trails and circuits; Hamilton paths and cycles
- Planar graphs and Euler's formula (introduction)
Counting and combinatorics
- Addition and multiplication principles; permutations and combinations
- Binomial coefficients and combinatorial identities
- Inclusion–exclusion principle
- Generating functions (introduction)
- Recurrence relations and characteristic equations (introduction)
Graph algorithms and structures
- Breadth-first and depth-first search
- Shortest paths: Dijkstra's algorithm (introduction)
- Matchings and Hall's marriage theorem (statement)
- Vertex and edge colorings; chromatic number
- Ramsey theory (introduction)
STEM / applied
Networks and optimization
- Network flows and max-flow min-cut theorem (overview)
- Traveling salesman problem and heuristic methods
- Scheduling and assignment problems modeled as graphs
- Social network analysis: centrality and clustering (introduction)
- Coding theory and Hamming graphs (optional application)
Computational combinatorics
- Implementation of graph algorithms in code
- Enumeration with backtracking and pruning
- Probabilistic method (introduction)
- Applications to computer science: trees in data structures
- Design of experiments and block designs (optional)
Notes
Topics reflect common graph theory and combinatorics syllabi at US colleges and universities. Some programs split graph theory and combinatorics across two courses.