Planar Graphs and Graph Coloring

Planar graphs and graph coloring are fundamental concepts in graph theory, a branch of mathematics that studies the properties and applications of graphs. A planar graph can be drawn on a plane without any edges crossing.

While graph coloring involves assigning colors to vertices such that no two adjacent vertices share the same color. These concepts have significant applications in various fields, including computer science, engineering, and geography.

Planar Graphs

A planar graph is a graph that can be embedded in the plane such that no edges intersect except at their endpoints. In other words, it can be drawn on a flat surface without any edges crossing.

Properties of Planar Graphs

• Euler's Formula: For any connected planar graph, the relationship between the number of:
  • Vertices (V)
  • Edges (E)
  • Faces (F) is given by : V − E + F = 2

• Kuratowski's Theorem : A finite graph is planar (i.e., it can be drawn on a plane without any edges crossing) if and only if it does not contain a subgraph that is a subdivision of either:
  • K5​ : The complete graph on 5 vertices, or
  • K3,3: The complete bipartite graph with partitions of 3 vertices each (also called the utility graph).

Graph Coloring

Graph coloring is the assignment of colors to vertices of a graph such that no two adjacent vertices share the same color. The minimum number of colors required to color a graph is called its chromatic number.

Types of Graph Coloring

• Vertex Coloring: Assigning colors to vertices so that no two adjacent vertices have the same color.
• Edge Coloring: Assigning colors to edges so that no two edges sharing the same vertex have the same color.
• Face Coloring: Assigning colors to faces of a planar graph so that no two faces sharing a boundary have the same color.

Applications in Engineering

Frequency Assignment

Graph coloring is used in assigning frequencies to radio stations, cell towers, and other communication devices to avoid interference.

Example: Assigning different frequencies to cell towers so that adjacent towers do not interfere with each other.

Scheduling

Graph coloring is used to schedule tasks, exams, and other events to ensure that no two conflicting tasks occur simultaneously.

Example: Scheduling exams for different courses so that students do not have overlapping exams.

Map Coloring

Face coloring of planar graphs is used in coloring maps so that no two adjacent regions share the same color.

Example: Coloring a political map where no two neighboring countries share the same color.

Register Allocation

In computer science, graph coloring is used for register allocation in compilers to efficiently use CPU registers.

Example: Allocating registers to variables in a program such that no two variables that are simultaneously live share the same register.

Circuit Design

Graph coloring is used in circuit design to minimize the number of layers required for connecting components on a printed circuit board.

Solved Examples - Planar Graphs and Graph Coloring

Example 1: Determine if the following graph is planar and find its chromatic number.

Graph: K4 (Complete graph with 4 vertices)

Solution:

K₄​ is planar because it can be drawn without any edges crossing.

Chromatic number: 4 (each vertex connects to every other vertex).

Example 2: Determine if K_3,3​ is planar.

Solution:

K_3,3​ is not planar as it cannot be drawn without edges crossing (Kuratowski's Theorem).

Example 3: Given a planar graph with 6 vertices and 9 edges, find the number of faces.

Solution:

Using Euler's formula:

V−E+F=2  ⟹  6−9+F=2  ⟹  F=5

Example 4: Find the chromatic number of a cycle graph with an odd number of vertices, say 5.

Solution:

For an odd cycle, the chromatic number is 3.

Example 5: Find the chromatic number of a bipartite graph.

Solution:

A bipartite graph is 2-colorable.

Example 6: Determine if the Petersen graph is planar.

Solution:

The Petersen graph is not planar.

Example 7: Find the chromatic number of the following planar graph: C5∪C3.

Solution:

Each cycle has a chromatic number of 3, so the combined graph has a chromatic number of 3.

Example 8: Verify Euler's formula for a graph with 8 vertices, 12 edges, and 6 faces.

Solution:

8−12+6=2(Holds true)

Example 9: Determine if the following graph is planar and find its chromatic number. Graph: K_2,3

Solution:

K_2,3 is planar.

Chromatic number: 2 (bipartite graph).

Practice Problems - Planar Graphs and Graph Coloring

• Determine if K₄,​ is planar and find its chromatic number.
• Given a planar graph with 10 vertices and 15 edges, find the number of faces.
• Find the chromatic number of a star graph Sn with n leaves.
• Prove that a tree is 2-colorable.
• Given a planar graph with 12 vertices, 18 edges, determine if Euler's formula holds.
• Find the chromatic number of a wheel graph W_n with n spokes.
• Determine if the dodecahedron graph is planar.
• Given a graph G with vertices V and E edges, if G is planar and E=3V−6, find the number of faces.
• Find the chromatic number of a complete bipartite graph K _m,n.
• Determine if the following graph is planar and find its chromatic number: Graph: C₄∪K₂

Related Articles

Four Color Theorem and Kuratowski’s Theorem in Discrete Mathematics

Fundamentals of Graph Theory

GATE CS Corner Questions

Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them.

• GATE CS 2005 Question 10
• GATE CS 2005, Question 47
• GATE CS 2004, Question 77
• GATE CS 2002, Question 4
• GATE CS 2015 Set-1, Question 63
• GATE CS 2008, Question 3
• GATE CS 2016 Set-2, Question 13
  

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