12.4 Navigating Graphs

Graph traversals and coloring are key concepts in graph theory. They help us understand how to navigate and organize complex networks of information. These techniques have practical applications in scheduling, resource allocation, and problem-solving across various fields.

Walks, trails, and paths describe different ways to move through a graph. Graph coloring assigns colors to vertices, ensuring no adjacent ones share a color. These ideas are crucial for optimizing schedules, allocating resources, and solving real-world problems efficiently.

Graph Traversals

Walks, trails, and paths

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• Walk represents a sequence of vertices and edges that begins and ends with vertices
  • Allows repetition of vertices and edges (can revisit nodes and retrace steps)
  • Closed walk (circuit) starts and ends at the same vertex (round trip)
• Trail is a walk in which no edge is repeated
  • Vertices may be repeated (can revisit nodes but not retrace steps)
  • Closed trail forms a circuit (round trip without retracing steps)
• Path is a walk in which no vertex or edge is repeated
  • Most restrictive traversal (visit each node at most once)
  • Closed path forms a cycle (round trip visiting each node once)
  • An Euler path is a path that visits every edge exactly once

Graph Types and Properties

• Connectivity refers to how well-connected a graph is
  • A graph is connected if there is a path between any two vertices
• Degree of a vertex is the number of edges connected to it
• Directed graph (digraph) has edges with a specific direction
• Weighted graph assigns values or weights to edges

Graph Coloring

Graph coloring techniques

• Graph coloring involves assigning colors to vertices of a graph
  • Adjacent vertices must have different colors (no neighboring nodes can share a color)
• Applies to various scheduling and resource allocation problems
  • Scheduling: vertices represent tasks or events, edges represent conflicts or constraints, colors represent time slots or resources
    • Course timetabling (assign time slots to classes avoiding conflicts)
    • Exam scheduling (assign exam slots to courses avoiding overlaps)
  • Resource allocation: vertices represent objects or entities, edges represent compatibility or conflict, colors represent resources or categories
    • Radio frequency assignment (assign frequencies to stations avoiding interference)
    • Map coloring (assign colors to regions avoiding adjacent regions with the same color)

Chromatic number significance

• Chromatic number $\chi(G)$ represents the minimum number of colors needed to color a graph $G$
  • Ensures no two adjacent vertices have the same color (proper coloring)
• Determines the minimum resources required in practical applications
  • Time slots, frequencies, or channels needed in scheduling problems
    • Minimum time slots to schedule all tasks without conflicts
    • Minimum frequencies to assign to stations without interference
  • Categories or types needed in resource allocation problems
    • Minimum colors to distinguish all regions on a map
    • Minimum types of resources to allocate to all entities without conflicts
• Lower chromatic number indicates more efficient resource utilization
  • Fewer colors needed, more optimized solution (fewer time slots, frequencies, or categories required)
• Higher chromatic number suggests more complex constraints or conflicts
  • More colors needed, less optimized solution (more time slots, frequencies, or categories required)
• A Hamilton path, which visits every vertex exactly once, can be useful in determining the chromatic number