DISCRETE MATHEMATICS 2
List of theoretical topics for the final exam1. Graphs - basic definitions.
2. Trees – definitions and properties (with proofs).
3. Problem of finding optimal spanning trees – Kruskal’s Algorithm (with proof of correctness).
4. Connectivity. Inequality relating connectivity,edge connectivity and minimum degree (Whitney's theorem with proof).
5. Euler tours and trails. Euler’s Theorem and corollary (with proof). Finding Euler tours. Fleury’s algorithm.
6. Hamilton paths and cycles. Necessary condition for existence of a Hamilton cycle in a graph (with proof). Ore’s Theorem (with proof). Dirac’s Theorem.
7. The Chinese Postman Problem. The Travellling Salesmen Problem.
8. Bipartite graphs. Characterization of bipartite graphs (with proof). Matchings.
9. Colorings of edges, the chromatic index. Vizing’s Theorem.
10. Coloring of vertices, the chromatic number, critical graphs, applications of coloring problems in scheduling and planning. Brooks’ Theorem.
11. Planar graphs. Dual graphs. Kuratowski’s Theorem. Euler’s Formula (with proof) and corollaries (with proofs). The Four Colour Conjecture. Heawood’s Theorem (with proof).
12. König’s Theorem (with proof). Systems of distinct representatives, Hall’s Theorem (with proof). Hall’s Theorem for bipartite graphs (with proof).
13. Flows in networks. Basic definitions. Problem of finding maximum flow in a network. Ford-Fulkerson’s Theorem.
