Graph Theory: Planar Graphs and Euler Circuits

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Questions and Answers

What is a planar graph?

A graph where edges can cross each other

A graph that can be drawn without edges crossing (correct)

A graph with no vertices

A graph that includes a Hamiltonian circuit

A graph is considered planar if at least one of its edges crosses another.

False

What term is used to describe a graph whose edges and vertices come from a given graph that is not planar?

subgraph

If a graph can be redrawn without any __________ between edges, it is classified as a planar graph.

crossing Signup and view all the answers

Match the following terms with their definitions:

Planar Graph = A graph without intersecting edges Hamiltonian Circuit = A route that visits each vertex exactly once and returns to the starting point Subgraph = A graph formed from a subset of a given graph's vertices and edges Redrawing = The process of altering the layout of a graph to improve clarity Signup and view all the answers

Which of the following statements about the Utilities Graph is true?

It contains a Hamiltonian circuit. Signup and view all the answers

Redrawing edges such that they do not cross is always possible for any graph.

False Signup and view all the answers

What main problem arises when trying to create a planar drawing of the Utilities Graph?

Edges cross each other. Signup and view all the answers

Which of the following statements is true about an Euler circuit?

It begins and ends at the same vertex and uses each edge exactly once. Signup and view all the answers

An Euler circuit can exist in a graph if at least one vertex has an odd degree.

False Signup and view all the answers

What is the term used for the number of edges that meet at a vertex?

degree Signup and view all the answers

A circuit that uses every edge exactly once is called an ______ circuit.

Euler Signup and view all the answers

In order for a graph to be Eulerian, which condition must be satisfied?

All vertices must have an even degree. Signup and view all the answers

The path A–B–C–E–H–G–E–B–D–A is an Euler circuit.

False Signup and view all the answers

How can Euler prove that a graph cannot have an Euler circuit?

By examining the number of edges meeting at each vertex. Signup and view all the answers

What time will it be 8 hours after 9 o’clock on a 12-hour clock?

5 o’clock Signup and view all the answers

What is the minimum number of colors required to color the graph represented by the fictional map according to the content?

4 Signup and view all the answers

A graph can be colored using only two colors if it has a ring of five vertices.

False Signup and view all the answers

If the time now is 10 o'clock, then 7 hours ago it was 3 o'clock.

True Signup and view all the answers

What day of the week is 16 days after Monday?

Wednesday Signup and view all the answers

What is the term used to describe a graph that can be colored with four colors?

4-colorable Signup and view all the answers

If today is Friday, then 14 days from now will also be _____

Friday Signup and view all the answers

The four-color theorem states that every planar graph can be colored with ______ colors.

four Signup and view all the answers

Match the following terms related to graph coloring with their definitions:

Planar graph = A graph that can be drawn on a plane without edges crossing Vertex = A point where two or more edges meet Edge = A line connecting two vertices Graph coloring = The assignment of colors to the vertices of a graph such that no two adjacent vertices share the same color Signup and view all the answers

Match the day with its corresponding number in modular arithmetic:

Friday = 0 Saturday = 1 Sunday = 2 Monday = 3 Tuesday = 4 Wednesday = 5 Thursday = 6 Signup and view all the answers

Is the congruence 29 ≡ 8 mod 3 true or false?

True Signup and view all the answers

In the fictional map example, how is connectivity between neighboring countries represented in the graph?

By edges Signup and view all the answers

According to the four-color theorem, it is possible for a planar graph to require more than four colors.

False Signup and view all the answers

The remainder of 16 divided by 7 is 2.

True Signup and view all the answers

What strategy can be used to color the graph vertices effectively?

Start with one vertex, assign it a color, and then assign colors to adjacent vertices while minimizing the number of new colors used. Signup and view all the answers

What was the day of the week on July 4, 2017?

Tuesday Signup and view all the answers

What day of the week is July 4, 2022?

Monday Signup and view all the answers

The year 2020 is not a leap year.

False Signup and view all the answers

What is the result of $61 \mod 12$?

1 Signup and view all the answers

To find what time it was 57 hours ago from 5 o’clock, calculate $(5 - 57) \mod ______$.

12 Signup and view all the answers

Match the operations with their results:

(33 - 16) mod 6 = 5 (14 - 27) mod 5 = 2 (23 + 38) mod 12 = 1 (5 - 57) mod 12 = 8 Signup and view all the answers

What does the expression 1826 modulo 7 represent?

The remainder when 1826 is divided by 7 Signup and view all the answers

In modular arithmetic, the result of an operation mod n is always less than n.

True Signup and view all the answers

How many extra days are counted in a leap year?

1 Signup and view all the answers

What is the result of the expression (15 * 23) mod 11?

4 Signup and view all the answers

The check digit of an ISBN is calculated using a multiplication modulo system.

False Signup and view all the answers

How many digits are there in a standard ISBN number?

13 Signup and view all the answers

If an integer x is 5, then the expression (x - 57) mod 12 results in __.

8 Signup and view all the answers

Match the following ISBN components with their descriptions:

d1 = First digit of ISBN country code = Indicates the region or country of publication publisher number = Represents the publisher identifier check digit = Final digit that ensures the correctness of ISBN Signup and view all the answers

What is the purpose of the check digit in an ISBN?

To ensure accuracy in data entry Signup and view all the answers

Transposition errors are rarely caught by the ISBN coding system.

False Signup and view all the answers

What happens when an incorrect ISBN is sent?

The receiving clerk calculates the check digit to verify its correctness. Signup and view all the answers

What does each dot in a graph typically represent in social network connections?

A friend Signup and view all the answers

In a graph, all edges must begin and end at vertices.

True Signup and view all the answers

Who is the student that has only one study group in common with others?

Kayla Signup and view all the answers

Amber is involved in __________ study groups.

three Signup and view all the answers

Match the following students with their number of study groups:

Amber = 3 Kayla = 1 Laura = 2 Dan = 2 Signup and view all the answers

Which type of connection in a graph indicates that two elements share a common attribute?

Edge Signup and view all the answers

Multiple edges can exist between the same two vertices.

True Signup and view all the answers

How many study groups does Laura have in common with the others?

two Signup and view all the answers

What is the purpose of a check digit in ISBN and UPC codes?

To ensure the number is valid and to help catch errors Signup and view all the answers

The Luhn algorithm is used to validate credit card numbers based on modular arithmetic.

True Signup and view all the answers

What is the final sum required for the Luhn algorithm to determine a valid credit card number?

0 mod 10 Signup and view all the answers

The check digit for the ISBN of 'The Equation that Couldn't Be Solved' is _____

3 Signup and view all the answers

Match the following coding systems with their descriptions:

ISBN = Used primarily for books UPC = Used for grocery items Credit Card = Uses the Luhn algorithm for validation Bar Code = Represents product information in a machine-readable format Signup and view all the answers

Which of the following statements about UPC is NOT true?

UPC codes are used only for books Signup and view all the answers

UPC codes and ISBNs use the same method to calculate their check digits.

False Signup and view all the answers

What is the typical length of a credit card number?

13 to 16 digits Signup and view all the answers

Which statement about an Euler circuit is true?

Every edge must be traversed exactly once. Signup and view all the answers

For a graph to be Eulerian, at least one vertex must have an odd degree.

False Signup and view all the answers

How can one determine if a graph has an Euler circuit?

Check that all vertices have an even degree. Signup and view all the answers

A path that begins and ends at the same vertex is known as a ______.

circuit Signup and view all the answers

Which of the following is an example of an Euler circuit?

B–D–F–G–H–E–C–B–A Signup and view all the answers

An Euler circuit can exist in a graph if it has at least one vertex of even degree.

False Signup and view all the answers

What observation did Euler make regarding the degree of vertices for an Euler circuit to exist?

Each vertex must have an even degree. Signup and view all the answers

What is the result of calculating $15 mod 6$?

1 Signup and view all the answers

If today is Thursday, then 6 days from now it will be Thursday.

False Signup and view all the answers

How many hours ago was it if the current time is 2 o’clock and it was at 9 o’clock?

5 Signup and view all the answers

The result of $8 + 5$ on a 12-hour clock is _____

1 Signup and view all the answers

Match the days of the week with their corresponding numbers in modular arithmetic:

Monday = 0 Tuesday = 1 Wednesday = 2 Thursday = 3 Friday = 4 Saturday = 5 Sunday = 6 Signup and view all the answers

What day of the week is represented by the number 2 in modular arithmetic?

Wednesday Signup and view all the answers

The congruence statement $29  8$ mod 3 is true.

True Signup and view all the answers

What will be the time 7 hours ago if it is currently 9 o'clock?

2 Signup and view all the answers

What is the minimum number of colors required to color the fictional map described in the content?

4 Signup and view all the answers

The graph representing the fictional map is always 3-colorable.

False Signup and view all the answers

What technique is suggested for coloring the vertices of a graph efficiently?

Start with one vertex, color it, and then color connected vertices one by one. Signup and view all the answers

Match the following concepts with their descriptions:

Four-color theorem = Every planar graph can be colored with four colors Planar graph = A graph that can be drawn without edges crossing Graph coloring = Assigning colors to vertices without adjacent vertices sharing a color Neighboring countries = Countries that share a common boundary Signup and view all the answers

How is the connectivity between neighboring countries represented in the graph?

With edges connecting the vertices Signup and view all the answers

A graph can be colored with only three colors if it has a ring of five vertices.

False Signup and view all the answers

The vertices of the graph are colored to ensure that no two vertices connected by an edge share the same ______.

color Signup and view all the answers

What is the relationship described by Euler's formula for a planar graph?

$v + f = e + 2$ Signup and view all the answers

In graph theory, the region surrounding a graph is called the infinite face.

True Signup and view all the answers

How many faces are counted in a planar graph with seven edges and five vertices, according to Euler's formula?

4 Signup and view all the answers

In graph theory, which of the following is NOT a practical application of graph coloring?

Creating non-planar graphs Signup and view all the answers

What is the term used for a graph that can be drawn on a plane without edges overlapping?

Planar Graph Signup and view all the answers

According to the four-color theorem, a planar graph may require more than four colors for proper coloring.

False Signup and view all the answers

What is the minimum number of colors required to color the graph represented by the fictional map?

4 Signup and view all the answers

The four-color theorem guarantees that a planar graph can use only three colors.

False Signup and view all the answers

How does the connectivity between neighboring countries get represented in a graph?

By drawing edges between vertices corresponding to the countries. Signup and view all the answers

The graph in the example requires a minimum of _____ colors for proper vertex coloring.

four Signup and view all the answers

What is the total weight of the Hamiltonian circuit A-D-B-F-C-E-A?

36 Signup and view all the answers

The Hamiltonian circuit mentioned can only be traversed in one direction.

False Signup and view all the answers

What is the primary purpose of using weights in a graph?

To represent quantities such as distances or costs. Signup and view all the answers

The cost of traveling between cities can be represented as the weight of the _____ in a graph.

edge Signup and view all the answers

Match the following cities with the corresponding edge weights in the context of travel costs:

London = 160 Paris = 250 Madrid = 240 Rome = 200 Signup and view all the answers

Which algorithm is mentioned for finding a low-cost route visiting each city exactly once?

Greedy Algorithm Signup and view all the answers

The edge-picking algorithm guarantees an optimal route in all cases.

False Signup and view all the answers

What was the smallest weight edge traveled to from Paris in the example?

Madrid Signup and view all the answers

The route A–D–B–F–C–E–A completes a _____ circuit.

Hamiltonian Signup and view all the answers

What happens when marking edges during the algorithm?

They may either complete a circuit or add a third edge to a vertex. Signup and view all the answers

What condition must be met for a graph to contain an Euler circuit?

All vertices have even degree. Signup and view all the answers

An Euler path can exist in a graph with all vertices having even degrees.

False Signup and view all the answers

In the context of the Euler paths, which two cities had to be the start and end points for the photographer's trip?

Alameda and Dover Signup and view all the answers

Euler paths traverse every edge in a graph exactly ______.

once Signup and view all the answers

Match the following terms related to Euler circuits and paths with their definitions:

Euler Circuit = Starts and ends at the same vertex, uses every edge once. Euler Path = Uses every edge once, does not require returning to the start. Eulerian Graph = A graph where every vertex has an even degree. Odd Degree Vertex = A vertex with an odd number of connecting edges. Signup and view all the answers

Which of the following represents a valid Euler circuit for the given graph?

B–A–F–B–E–F–G–E–D–G–B–D–C–B Signup and view all the answers

There are multiple Euler circuits possible in an Eulerian graph.

True Signup and view all the answers

In a complete graph with six vertices, what can be said about the circuits found by algorithms?

They often outperform trial and error methods. Signup and view all the answers

The greedy algorithm selects the highest weight edge at each step.

False Signup and view all the answers

What is the total weight of the Hamiltonian circuit found using the greedy algorithm starting at vertex A?

42 Signup and view all the answers

The edge with the smallest weight in the example was __________.

BD Signup and view all the answers

Which edge was chosen from vertex E in the greedy algorithm example?

CE Signup and view all the answers

Match the algorithms with their characteristics:

Greedy Algorithm = Chooses the cheapest option at each step Edge-Picking Algorithm = Highlights edges based on weight Hamiltonian Circuit = Visits all vertices and returns to the start Complete Graph = A graph where every pair of vertices is connected Signup and view all the answers

In the edge-picking algorithm, edges with the same weight can be represented together.

True Signup and view all the answers

What is Euler's formula related to in the context of planar graphs?

The relationship between edges, vertices, and faces Signup and view all the answers

What is a characteristic feature of a complete graph?

Every pair of vertices is connected by an edge. Signup and view all the answers

In a planar graph, the exterior region is considered as one of the faces.

True Signup and view all the answers

What is the minimum number of colors required to color any planar graph according to the four-color theorem?

four Signup and view all the answers

What is the first edge selected when using the edge-picking algorithm in the weighted graph?

BD with weight 2 Signup and view all the answers

In Euler's formula, if a graph has 5 vertices and 4 faces, then it must have ______ edges.

7 Signup and view all the answers

Match the following terms with their definitions related to graph theory:

Vertices = Points in the graph where edges meet Edges = Lines connecting vertices Faces = Regions divided by edges in a planar graph Planar graph = A graph that can be drawn on a plane without crossing edges Signup and view all the answers

What does the connection between map coloring and graph theory allow for?

Efficient scheduling and task assignments Signup and view all the answers

A graph that can be colored with only two colors can contain cycles of any length.

False Signup and view all the answers

What happens when neighboring countries share a border in graph theory?

An edge is created between the corresponding vertices. Signup and view all the answers

The region surrounding the graph in a planar drawing is known as the ______ face.

infinite Signup and view all the answers

According to the nonplanar graph theorem, which of the following graphs cannot be contracted to K5?

Complete graph on five vertices (K5) Signup and view all the answers

What is required for a graph to be classified as Eulerian?

All vertices must have an even degree. Signup and view all the answers

An Euler path requires returning to the starting point.

False Signup and view all the answers

Name a city with an odd degree in the road example provided.

Alameda or Dover Signup and view all the answers

A path that visits every vertex exactly once is called a ______ circuit.

Hamiltonian Signup and view all the answers

How many vertices with odd degrees must exist for an Euler path to be possible?

Exactly two Signup and view all the answers

All Euler circuits must cover each edge of the graph exactly twice.

False Signup and view all the answers

What must a photographer do to successfully traverse all roads exactly once in the city map?

Start at one of the cities with an odd degree. Signup and view all the answers

An example of an odd degree vertex in the road map is ______.

Alameda or Dover Signup and view all the answers

What is a complete graph?

A connected graph where every possible edge is drawn between vertices Signup and view all the answers

A graph is considered connected if all vertices can be reached from any other vertex.

True Signup and view all the answers

What is the chromatic number of a planar graph according to the four-color theorem?

Four Signup and view all the answers

What do you call an edge that begins and ends at the same vertex?

loop Signup and view all the answers

A nonplanar graph always requires four or fewer colors to color it.

False Signup and view all the answers

A graph without any edges is considered __________.

not connected Signup and view all the answers

What is the minimum number of colors needed to color a graph so that no two connected vertices have the same color?

chromatic number Signup and view all the answers

Which of the following best describes equivalent graphs?

Graphs that represent the same connections between vertices but may differ in arrangement Signup and view all the answers

In scheduling meetings among clubs with shared members, the number of required time slots corresponds to the graph's __________ for proper coloring.

chromatic number Signup and view all the answers

What is the significance of the edges in a graph?

Edges represent connections between vertices. Signup and view all the answers

Match the graph type with its continuous coloring conditions:

Planar Graph = Can be colored with four colors Nonplanar Graph = May need more than four colors Two-colorable Graph = Can be colored with two colors Circuit Graph = Contains even number of vertices Signup and view all the answers

What is the configuration of edges, vertices, and faces in a planar graph that verifies Euler’s formula?

5 vertices, 4 faces, 7 edges Signup and view all the answers

In a planar graph, the infinite face is not considered a face.

False Signup and view all the answers

What is Euler's formula for a planar graph?

v + f = e + 2 Signup and view all the answers

A planar graph can be drawn in such a way that no edges __________.

cross Signup and view all the answers

Match each term with its corresponding definition:

Vertices = Points in the graph representing objects Edges = Lines connecting the vertices Faces = Regions bordered by edges in the graph Infinite Face = The region outside the graph Signup and view all the answers

How many colors are needed to color a map of countries, ensuring no two neighbors share the same color, as indicated by the content?

4 Signup and view all the answers

Graph coloring can be applied in various practical applications, including scheduling tasks.

True Signup and view all the answers

When counting the faces in a planar graph, what additional face must always be included?

the infinite face Signup and view all the answers

In graph theory, an edge connects two __________.

vertices Signup and view all the answers

A country is represented by a vertex in graph theory if it __________.

is placed anywhere within its own boundary Signup and view all the answers

The graph in Figure 5.23 is 4-colorable.

False Signup and view all the answers

A map can be colored using the fewest colors possible with __________ where no two adjacent countries share the same color.

vertices Signup and view all the answers

Match the following graph-coloring terms with their descriptions:

Vertex = A point representing a country in a graph Edge = A connection between two vertices Planar graph = A graph that can be drawn without edges crossing Graph coloring = Assigning colors to vertices so no two adjacent share the same color Signup and view all the answers

Which is true about the four-color theorem?

It guarantees that no planar graph will require more than four colors. Signup and view all the answers

To color a graph effectively, you can reuse colors when connecting non-adjacent vertices.

True Signup and view all the answers

If a graph requires fewer colors than the maximum, it is referred to as being __________.

colorable Signup and view all the answers

Why will a particular vertex need a fourth color in the example provided?

It is adjacent to a ring of five vertices. Signup and view all the answers

What is the remainder when the product (15 * 23) is divided by 11?

4 Signup and view all the answers

The check digit of an ISBN is calculated using an addition modulo system.

False Signup and view all the answers

If the current time is 5 o'clock, what time was it 57 hours ago?

8 o'clock Signup and view all the answers

The first three digits of an ISBN are always _____ or 979.

978 Signup and view all the answers

Match the following components of an ISBN with their descriptions:

d1 = First digit of the ISBN d13 = Check digit 978 = Prefix for standard books 9 digits = Identifies country, publisher, and title Signup and view all the answers

Which of the following statements correctly describes the traveling salesman problem?

It finds the shortest route to visit each city once and return to the starting point. Signup and view all the answers

The Hamiltonian circuit must visit each city exactly once.

True Signup and view all the answers

What is the total distance travelled by the route Chicago–New York–Dallas–Philadelphia–Atlanta–Washington, D.C.–Chicago?

5197 Signup and view all the answers

The two algorithms mentioned for finding a solution in complete graphs are the greedy algorithm and the __________ algorithm.

edge-picking Signup and view all the answers

Match the following routes with their total distances:

Chicago–New York–Dallas–Philadelphia–Atlanta–Washington, D.C.–Chicago = 5197 miles Chicago–Philadelphia–Dallas–Washington, D.C.–Atlanta–New York–Chicago = 5154 miles Chicago–Washington, D.C.–Dallas–New York–Atlanta–Philadelphia–Chicago = 5239 miles Signup and view all the answers

Which statement about complete graphs is true?

Every possible edge is drawn between vertices. Signup and view all the answers

Finding the optimal Hamiltonian circuit in a weighted graph has a known shortcut.

False Signup and view all the answers

What is the total distance for the route Chicago–Washington, D.C.–Dallas–New York–Atlanta–Philadelphia–Chicago?

5239 Signup and view all the answers

In the context of the traveling salesman problem, a route that visits each city only once is termed a __________ circuit.

Hamiltonian Signup and view all the answers

Which of the following statements accurately describes a Hamiltonian circuit?

It visits each vertex once and returns to the starting vertex. Signup and view all the answers

Dirac’s Theorem guarantees that a graph is Hamiltonian if every vertex has a degree of at least $n/2$.

True Signup and view all the answers

What does a weighted graph associate with each edge?

a weight Signup and view all the answers

A sequence of flights that starts and ends at the same city without visiting any city twice is represented by a Hamiltonian __________.

circuit Signup and view all the answers

Match the following theorems or concepts with their correct descriptions:

Hamiltonian Circuit = A path that visits all vertices exactly once, returning to the start. Weighted Graph = A graph in which edges have associated values. Dirac's Theorem = A theorem providing conditions for the existence of Hamiltonian circuits. Flight Graph = A graph representing flights between cities as vertices connected by edges. Signup and view all the answers

What is the significance of the weights on the edges of a weighted graph?

They symbolize distances or costs associated with the connections. Signup and view all the answers

A Hamiltonian circuit can exist in a graph with vertices of varying degree, as long as there is a sufficient number of edges.

False Signup and view all the answers

What is 7 hours before 3 o'clock on a 12-hour clock?

8 o'clock Signup and view all the answers

16 days from Friday will be Saturday.

False Signup and view all the answers

What day of the week is 14 days after Wednesday?

Wednesday Signup and view all the answers

Using modular arithmetic, 29 ≡ 8 mod ______.

3 Signup and view all the answers

Match the following operations with their results on a 12-hour clock:

10 + 5 = 3 8 - 9 = 7 11 + 4 = 3 2 - 5 = 9 Signup and view all the answers

Modular arithmetic can be used to represent day-of-the-week calculations.

True Signup and view all the answers

Calculate 15 - 8 mod 12.

7 Signup and view all the answers

If today is Monday, then 9 days from now will be a ______.

Wednesday Signup and view all the answers

What is a key characteristic of a planar graph?

No edges cross over each other. Signup and view all the answers

The Utilities Graph can be redrawn in a way that does not have any edge crossings.

False Signup and view all the answers

What is the term for a graph that is derived from another graph using its edges and vertices?

subgraph Signup and view all the answers

If a graph can be redrawn with no edges crossing, it is classified as a ______ graph.

planar Signup and view all the answers

Match each graph property with its correct description:

Planar Graph = A graph with no edges crossing Hamiltonian Graph = A graph containing a Hamiltonian circuit Subgraph = A graph formed from a portion of another graph Non-planar Graph = A graph requiring edge crossings in any configuration Signup and view all the answers

Which of the following statements about a subgraph is true?

A subgraph can have fewer vertices and edges than the original graph. Signup and view all the answers

An equivalent form of a non-planar graph can still be classified as non-planar if it has crossing edges.

True Signup and view all the answers

List one strategy to determine if a graph is not planar.

Finding a non-planar subgraph Signup and view all the answers

A graph that has edges which connect opposite vertices but require crossing is considered ______.

non-planar Signup and view all the answers

Which property makes the Utilities Graph non-planar?

It contains a Hamiltonian circuit. Signup and view all the answers

Study Notes

Introduction to Graphs

Graph theory is a branch of mathematics that analyzes connections.
Graphs are diagrams with vertices (points) and edges (lines or curves) connecting vertices.
Vertices represent entities, and edges illustrate relationships between them.
Graphs are used to model various connections, such as friendships on social media, roads between cities, and the internet.

Example 1 - Constructing a Graph

A table shows which students study together.
"X" indicates two students study in the same group.
Vertices represent students.
Edges connect vertices if corresponding students study together.

Example 1 - Solution

A graph shows how students are connected.
The student who participates in the most study groups is identified through the number of connections to other students.
The student with one connection is identified as someone who forms only one study group.

Introduction to Graphs (continued)

A graph can include vertices without edges or multiple edges connecting the same vertices; and an edge that is connected to the vertex (a loop).
A connected graph means any vertex can reach any other vertex by following edges.
In a complete graph, every possible edge is drawn between all vertices.
Graphs can represent various relationships and connections in real-world scenarios.

Euler Circuits

Euler circuits are paths that use every edge exactly once in a graph.
Euler circuits are called Eulerian.
The graph must have all even degrees of vertices.
The Königsberg bridges problem is solved by representing the land regions as vertices and connecting two vertices if a bridge exists between them.
Euler circuits are important in solving problems involving traversing edges exactly once.

Euler Paths

Euler paths are paths that use every edge exactly once (without returning to the starting point).
The graph must have exactly two vertices of odd degree.
The Euler path must start at one odd degree vertex and end at the other point of odd degree vertex.
Euler paths are used for route planning and traversing networks.

Weighted Graphs

A weighted graph has edges with values (weights), which can represent any quantity (e.g., distance, cost).
For Hamiltonian circuits in a weighted graph, summing the weights along the traversed edges yields the total distance traveled.
The traveling salesman problem seeks the shortest total distance for a Hamiltonian circuit, where the goal is to travel to all locations and return to the starting point in the shortest possible way.
Weighted graphs are used to model problems with various quantities associated with relationships.

Hamiltonian Circuits

A Hamiltonian circuit is a path that visits every vertex exactly once and returns to the starting vertex.
Dirac's theorem gives conditions for a graph to contain a Hamiltonian circuit: the connected graph must have at least 3 vertices, and every vertex has a degree equal to or greater than n/2, where n is the number of vertices.
The greedy algorithm and edge-picking algorithm are strategies for finding good Hamiltonian circuits. These methods can find a solution that might not be the absolute best but one that is "pretty good" or a "good solution" based on the given constraints.
Hamiltonian circuits are vital in problems like finding optimal routes or tours through a network of locations.

Planarity

A planar graph can be drawn on a plane so that no edges intersect.
The utilities graph puzzle is illustrated with three utility companies and three houses; if the pipes connect without crossing, the graph is planar; if they cross, the graph is nonplanar.
A subgraph of a graph consists of vertices and edges from the graph.
K5 is a complete graph with five vertices.
The subgraph theorem states that if a graph contains a non-planar subgraph (like the Utilities Graph or K5), the graph itself is non-planar.
Contractions in graphs provide a way to investigate nonplanar graphs and identify if such a graph is a subgraph of a larger graph.
Planar graphs are crucial in many applications involving physical layouts, like map designs.

Euler's Formula

Euler's formula relates the number of vertices (v), edges (e), and faces (f) of a planar graph via the relationship v + f = e + 2.

Graph Coloring

Graph coloring is used to color each vertex of a graph such that no two connected vertices share the same color.
The chromatic number is the minimum number of colors required to color a graph so that each edge connects vertices of different colors.
The four-color theorem states that every planar graph is 4-colorable, so only 4 colors are required to complete such a coloring.
Applications of graph coloring are widely varied and include scheduling problems; colorings are done so that when vertices are connected, they are not the same colors.
Graph coloring is used in various scenarios where conflicts need to be avoided, and scheduling or design problems require careful assignment of different resources to avoid simultaneous use of the same resource.

Modular Arithmetic

Modular arithmetic works with remainders when integers are divided.
Modular arithmetic is used for clock arithmetic and day-of-the-week calculations; 'modulo n' refers to the repeated pattern of mathematical results based on the remainder after division.
The concept of modulus n provides a mathematical framework for phenomena that repeat in cycles or involve cyclic patterns in mathematics.

Mathematical Systems

Mathematical systems describe and model how a set of elements behaves with certain operations.
Modular arithmetic is a specific system with certain rules for addition and subtractions, resulting in predictable patterns that are fundamental to many aspects of applied mathematics.
Mathematical systems provide structure and models for various forms of data/information analysis, problem solving, and pattern recognition.

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Description

This quiz covers key concepts in graph theory, focusing on planar graphs and Euler circuits. You will explore definitions, properties, and conditions related to these types of graphs. Test your understanding of the Utilities Graph and related principles.

Graph Theory: Planar Graphs and Euler Circuits

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Questions and Answers

What is a planar graph?

A graph is considered planar if at least one of its edges crosses another.

What term is used to describe a graph whose edges and vertices come from a given graph that is not planar?

If a graph can be redrawn without any __________ between edges, it is classified as a planar graph.

Match the following terms with their definitions:

Which of the following statements about the Utilities Graph is true?

Redrawing edges such that they do not cross is always possible for any graph.

What main problem arises when trying to create a planar drawing of the Utilities Graph?

Which of the following statements is true about an Euler circuit?

An Euler circuit can exist in a graph if at least one vertex has an odd degree.

What is the term used for the number of edges that meet at a vertex?

A circuit that uses every edge exactly once is called an ______ circuit.

In order for a graph to be Eulerian, which condition must be satisfied?

The path A–B–C–E–H–G–E–B–D–A is an Euler circuit.

How can Euler prove that a graph cannot have an Euler circuit?

What time will it be 8 hours after 9 o’clock on a 12-hour clock?

What is the minimum number of colors required to color the graph represented by the fictional map according to the content?

A graph can be colored using only two colors if it has a ring of five vertices.

If the time now is 10 o'clock, then 7 hours ago it was 3 o'clock.

What day of the week is 16 days after Monday?

What is the term used to describe a graph that can be colored with four colors?

If today is Friday, then 14 days from now will also be _____

The four-color theorem states that every planar graph can be colored with ______ colors.

Match the following terms related to graph coloring with their definitions:

Match the day with its corresponding number in modular arithmetic:

Is the congruence 29 ≡ 8 mod 3 true or false?

In the fictional map example, how is connectivity between neighboring countries represented in the graph?

According to the four-color theorem, it is possible for a planar graph to require more than four colors.

The remainder of 16 divided by 7 is 2.

What strategy can be used to color the graph vertices effectively?

What was the day of the week on July 4, 2017?

What day of the week is July 4, 2022?

The year 2020 is not a leap year.

What is the result of $61 \mod 12$?

To find what time it was 57 hours ago from 5 o’clock, calculate $(5 - 57) \mod ______$.

Match the operations with their results:

What does the expression 1826 modulo 7 represent?

In modular arithmetic, the result of an operation mod n is always less than n.

How many extra days are counted in a leap year?

What is the result of the expression (15 * 23) mod 11?

The check digit of an ISBN is calculated using a multiplication modulo system.

How many digits are there in a standard ISBN number?

If an integer x is 5, then the expression (x - 57) mod 12 results in __.

Match the following ISBN components with their descriptions:

What is the purpose of the check digit in an ISBN?

Transposition errors are rarely caught by the ISBN coding system.

What happens when an incorrect ISBN is sent?

What does each dot in a graph typically represent in social network connections?

In a graph, all edges must begin and end at vertices.

Who is the student that has only one study group in common with others?

Amber is involved in __________ study groups.

Match the following students with their number of study groups:

Which type of connection in a graph indicates that two elements share a common attribute?

Multiple edges can exist between the same two vertices.

How many study groups does Laura have in common with the others?

What is the purpose of a check digit in ISBN and UPC codes?

The Luhn algorithm is used to validate credit card numbers based on modular arithmetic.

What is the final sum required for the Luhn algorithm to determine a valid credit card number?

The check digit for the ISBN of 'The Equation that Couldn't Be Solved' is _____

Match the following coding systems with their descriptions:

Which of the following statements about UPC is NOT true?

UPC codes and ISBNs use the same method to calculate their check digits.

What is the typical length of a credit card number?

Which statement about an Euler circuit is true?

For a graph to be Eulerian, at least one vertex must have an odd degree.

How can one determine if a graph has an Euler circuit?

A path that begins and ends at the same vertex is known as a ______.

Which of the following is an example of an Euler circuit?

An Euler circuit can exist in a graph if it has at least one vertex of even degree.

What observation did Euler make regarding the degree of vertices for an Euler circuit to exist?

What is the result of calculating $15 mod 6$?

If today is Thursday, then 6 days from now it will be Thursday.

How many hours ago was it if the current time is 2 o’clock and it was at 9 o’clock?

The result of $8 + 5$ on a 12-hour clock is _____

Match the days of the week with their corresponding numbers in modular arithmetic:

What day of the week is represented by the number 2 in modular arithmetic?

What is the result of calculating $15 mod 6$?