Study Guide & Educational Overview
### Key Discrete Math Concepts:
* **Graph Traversals**: An Eulerian path visits every edge exactly once (requires 0 or 2 vertices of odd degree); a Hamiltonian path visits every vertex exactly once.
* **Trees & Spanning Networks**: A tree on n vertices has exactly n - 1 edges and no cycles; Kruskal's and Prim's algorithms construct minimum spanning trees (MST) on weighted graphs.
* **Combinatorics & Sets**: Pigeonhole Principle (placing n + 1 items into n containers guarantees an overlap); cardinality of power set P(S) = 2^n; inclusion-exclusion principle |A ∪ B| = |A| + |B| - |A ∩ B|.
* **Planar Graphs & Euler's Formula**: V - E + F = 2 holds for all connected planar graphs; the Four Color Theorem guarantees any planar map can be colored with at most 4 colors.
* **Graph Traversals**: An Eulerian path visits every edge exactly once (requires 0 or 2 vertices of odd degree); a Hamiltonian path visits every vertex exactly once.
* **Trees & Spanning Networks**: A tree on n vertices has exactly n - 1 edges and no cycles; Kruskal's and Prim's algorithms construct minimum spanning trees (MST) on weighted graphs.
* **Combinatorics & Sets**: Pigeonhole Principle (placing n + 1 items into n containers guarantees an overlap); cardinality of power set P(S) = 2^n; inclusion-exclusion principle |A ∪ B| = |A| + |B| - |A ∩ B|.
* **Planar Graphs & Euler's Formula**: V - E + F = 2 holds for all connected planar graphs; the Four Color Theorem guarantees any planar map can be colored with at most 4 colors.
Frequently Asked Questions
Leonhard Euler proved in 1736 that walking across all seven town bridges without retracing any was impossible because all four landmasses had an odd number of connecting bridges.
Permutations consider the order of arrangement (n! / (n-k)!), whereas combinations ignore order and select groups (n! / (k!(n-k)!)).
