Some NP-complete Problems in Graph Theory

Prof. Sin-Min Lee

Graph Theory

•An independent set is a subset S of the verticies of the graph, with no elements of S connected by an arc of the graph.

Coloring

• How do you assign a color to each vertex so that adjacent vertices are colored differently?

• Chromatic number of certain types of graphs.• k-Coloring is NP Complete.• Edge coloring

Planarity and Embeddings

K4 is planar

K5 is not

Euler’s formulaKuratowski’s theoremPlanarity algorithms

Flows and Matchings

• Meneger’s theorem (separating vertices)• Hall’s theorem (when is there a matching?)• Stable matchings• Various extensions and similar problems• Algorithms

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3

6

1

72

4

9

3

1 5

girls boys

BB: III – maybe two weeks?

AG: CH. 4 and 5.

Random Graphs

• Form probability spaces containing graphs or sequences of graphs as points.

• Simple properties of almost all graphs.

• Phase transition: as you add edges component size jumps from log(n) to cn.

Algebraic Graph Theory

• Cayley diagrams

• Adjacency and Laplacian Matrices their eigenvalues and the structure of various classes of graphs

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groupelements

generators

Algorithms• DFS, BFS, Dijkstra’s Algorithm...• Maximal Spanning Tree...• Planarity testing, drawing...• Max flow...• Finding matchings...• Finding paths and circuits...• Traveling salesperson algorithms...• Coloring algorithms...