StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q3E

Expert-verifiedFound in: Page 733

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

** Construct the dual graph for the map shown:**

**Then find the number of colors needed to color the map so that no two adjacent regions have the same color.**

The dual graph for the map given is:

And the number of colors needed to color the given map so that no two adjacent regions have the same color is \(3\).

In the question, a map is given.

**According to the definition of dual graph, each map can be represented by a graph and in every graph each region of the map is represented by a vertex also two vertices are connected by a edge if and only if the region represented by these vertices have a common border, then the obtained graph is known as dual graph.**

Construct the dual graph of the given map by determining one vertex in each region as follows:

Since, the vertex \(a\) is connected to all the other vertices, thus the color of the vertex \(a\) should be unique or different from all the other vertices and the vertices \(b\) and \(d\) are not directly connected to each other similarly vertices \(f,c\) and \(e\) are not directly connected to each other.

Therefore,

Region | Vertex | Color |

A | a | Blue |

B | b | Green |

C | c | Red |

D | d | Green |

E | e | Red |

F | f | Red |

Color the given map as shown:

Hence, \[3\] color are needed to color the given map.

94% of StudySmarter users get better grades.

Sign up for free