Exploring Planar Graph Coloring

What is a planar graph?

• A graph that can only be drawn in three dimensions.

• A graph with all edges crossing each other.

• A graph that can be drawn on a plane without edges crossing.

• A graph that cannot be represented visually.

Define graph coloring in the context of planar graphs.

• Graph coloring allows adjacent vertices to share the same color if they are not connected directly.

• Graph coloring for planar graphs is the assignment of colors to vertices so that no two adjacent vertices have the same color, with a maximum of four colors needed.

• Graph coloring for planar graphs requires at least five colors for any configuration.

• Graph coloring involves coloring edges instead of vertices in planar graphs.

What is the Four Color Theorem?

• The Four Color Theorem states that four colors are sufficient to color any map such that no adjacent regions share the same color.

• The Four Color Theorem states that three colors are sufficient to color any map.

• The Four Color Theorem applies only to maps of the United States.

• The Four Color Theorem requires five colors to ensure no adjacent regions share the same color.

How does graph coloring apply to scheduling problems?

• Graph coloring helps in scheduling by modeling tasks as vertices and conflicts as edges, minimizing colors to optimize time slots or resources.

• Graph coloring is used to determine the shortest path in a network.

• Graph coloring helps in identifying the most efficient algorithm for sorting data.

• Graph coloring is primarily concerned with optimizing database queries.

What is the chromatic number of a planar graph?

• 2

• 4

• 3

• 5

Explain the significance of bipartite graphs in graph coloring.

• Bipartite graphs cannot be colored at all.

• Bipartite graphs can be colored with two colors, simplifying graph coloring problems.

• Bipartite graphs are only useful in network flow problems.

• Bipartite graphs require three colors for proper coloring.

What is a proper coloring of a graph?

• A proper coloring assigns the same color to all vertices.

• A proper coloring involves coloring edges instead of vertices.

• A proper coloring of a graph is an assignment of colors to vertices such that no two adjacent vertices have the same color.

• A proper coloring is a method to connect vertices with edges.

How can you determine if a graph is planar?

• A graph is planar if it does not contain a subgraph that is a subdivision of K5 or K3,3.

• A graph is planar if it contains at least one vertex of degree three.

• A graph is planar if it has no cycles.

• A graph is planar if it can be drawn without any edges crossing.

What are some applications of graph coloring in real life?

• Designing logos for brands

• Painting walls in homes

• Applications of graph coloring include scheduling problems, register allocation in compilers, frequency assignment in mobile networks, and map coloring.

• Coloring books for children

Describe an algorithm used for coloring a graph.

• Depth-First Search Algorithm

• Dijkstra's Algorithm

• Breadth-First Search Algorithm

• Greedy Coloring Algorithm