18. Design and Analysis of Algorithms
Graphs are crucial structures used to represent information in problems such as map coloring and airline routing. By modeling states as vertices and their connections as edges, complex problems can be simplified, focusing on essential relationships while discarding irrelevant details. The concept of graph coloring illustrates the need to differentiate connected entities using minimal colors, which leads to significant mathematical insights.
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What we have learnt
- Graphs consist of vertices (nodes) and edges connecting them.
- The graph coloring problem ensures that connected vertices do not share the same color.
- The Four Color Theorem asserts that four colors are sufficient for any planar graph representation derived from a map.
Key Concepts
- -- Graph
- A graph is a collection of vertices and edges, where edges connect pairs of vertices.
- -- Vertex
- A vertex (or node) is a fundamental unit of a graph, representing an entity such as a state or city.
- -- Edge
- An edge is a connection between two vertices that may represent a relationship such as adjacency or routing.
- -- Graph Coloring
- The process of assigning colors to the vertices of a graph such that no two adjacent vertices share the same color.
- -- Four Color Theorem
- A theorem stating that four colors are sufficient to color any map such that no two adjacent regions share the same color.
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