# Graph Theory for Primary School Students with High Skills in Mathematics

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Settings

- Initial challenge (Adaptation): Activity posed as a challenge aimed to create curiosity in the students so that they want to learn about the subject.
- Practice of the new concept (Structuring): We use graphs to represent routes or pathways under certain conditions.
- Abstraction (Abstraction): Graphs are used to solve a problem of greater difficulty than previous ones, where more abstract mathematical processes appear, in this case induction and generalization.
- Closing activities (Reasoning): The objective is to relate the graphs that were studied with other mathematical concepts.

#### 2.2. Description of the Participants

## 3. Results

#### 3.1. Map Coloring

#### 3.2. Star Polygons

#### 3.3. Give Me Five

#### 3.4. Path Tracing/Eulerian Cycles

#### 3.5. Seven Bridges of Köningsberg

#### 3.6. Chess Routes

#### 3.7. Mathematical Challenges

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- How many times 2 people high-five when they meet?
- How many times 3 people high-five when they meet?
- How many times 4 people high-five when they meet?
- How many times 5 people high-five when they meet?
- How many times 9 people high-five when they meet?
- How many times 17 people high-five when they meet?
- How many times 27 people high-five when they meet?
- How did you get the answers?

- Can you find a way walking through the 4 areas of the city, crossing each bridge only once and going back to the starting point?
- Can you see any difference between the starting point and the other ones?
- Count the number of edges leaving out or arriving at each vertex.
- How many times would it be necessary to cross every bridge to complete a closed path going through all the bridges (at least once).

- Can the knight go through all the squares, passing only once through each one? Can you prove it?
- Now choose the chess piece you prefer and try again.
- These ways described by chess pieces are Hamiltonian paths. A Hamiltonian path is a path (nonclosed path) passing only once for every vertex.

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**Figure 1.**Basic graphs. Examples of (

**a**) graph; (

**b**) planar graph; (

**c**) nonplanar graph; (

**d**) Eulerian graph; (

**e**) non-Eulerian graph; (

**f**) Eulerian graph, but non-Hamiltonian graph; (

**g**) Hamiltonian graph, non-Eulerian graph.

**Figure 2.**First attempt of coloring the map of Spain. Drawing made by participant: (

**a**) A3; (

**b**) A6; (

**c**) A4.

**Figure 3.**First attempt of coloring the map of Spain. Drawing made by participant: (

**a**) A1; (

**b**) A2 (orange is yellow); (

**c**) A5; (

**d**) A7.

**Figure 6.**Star polygons. Drawing made by participant: (

**a**) A2 (similar to the one made by A1); (

**b**) A3.

**Figure 10.**Tracing Eulerian cycles. Drawing made by participant: (

**a**) A3; (

**b**) A5; (

**c**) A4; (

**d**) A6; (

**e**) A7.

**Figure 11.**Köningsberg bridges. Drawing made by participant: (

**a**) A7; (

**b**) A6; (

**c**) A5; (

**d**) A3; (

**e**) A4.

**Figure 12.**Graphs associated with Köningsberg bridges. Drawing made by participant: (

**a**) A6; (

**b**) A7; (

**c**) A5; (

**d**) A3; (

**e**) A4.

**Figure 13.**Chess routes with a knight. Drawing made by participant: (

**a**) A5; (

**b**) A7; (

**c**) A3; (

**d**) A4; (

**e**) A6.

**Figure 14.**Chess routes. Drawing made by participant: (

**a**) A5 using a rook; (

**b**) A3 using a rook; (

**c**) A7 using the queen; (

**d**) A4 using a rook; (

**e**) A6 using the king.

Stage | Activities | Topics | Session |
---|---|---|---|

Adaptation | Coloring of maps | Four color theorem | 1 |

Idea of graph and cycle | |||

Structuring | Köningsberg bridges problem | Graph associated, degree of a vertex, closed path | 2 |

Eulerian cycles | 3 | ||

Abstraction | Give me five | Complete graphs | 4 |

Induction | |||

Reasoning | Star polygons | Drawing star polygons | 5 |

Stage | Activities | Topics | Session |
---|---|---|---|

Adaptation | Coloring of maps | Four color theorem | 1 |

Idea of graph and cycle | |||

Structuring | Star polygons | Drawing star polygons | 2 |

Abstraction | Give me five | Complete graphs | 3 |

Induction | |||

Path tracing | Eulerian cycle, degree of a vertex | 4 | |

Reasoning | Köningsberg bridges problem | Closed path, graph associated | 5 |

Degree of a vertex | |||

Chess routes | Hamiltonian path | 6 |

Student | Age | Gender | Group |
---|---|---|---|

A1, A2 | 6 | Female | G1 |

A3 | 9 | Male | G2 |

A4, A5 | 10 | Female | G2 |

A6, A7 | 11 | Female | G3 |

Challenge | Group | Problem | Strategy | |
---|---|---|---|---|

1 | G1 | Set the minimum number of colors needed to color Castilla-La Mancha [25], such that neighboring regions do not have the same color | Simplification Simpler problem | |

G2 | Inside a regular pentagon, draw a five-pointed star without lifting the pencil from the paper | Simpler problem | ||

G3 | Draw two different star polygons inside a regular polygon of 17 edges | Simpler problem | ||

2 | G1 | Draw a cycle in a set of 20 nonaligned points, joining all points | Simpler problem | |

G2 | Inside an enneagon draw as many star polygons as you can | Simpler problem | ||

G3 | Find the Cartesian coordinates of 5 points (they pose some computations). Afterward, draw a star polygon and find all of their symmetry axes | Related problem | ||

3 | G1 | We are 15 friends playing with the ball, but we cannot pass the ball twice to the same classmate. How many times did we pass the ball? (They solve it by counting.) | Variation of the problem | |

G2 | Plan a trip through several places without passing twice by the same path | Variation of the problem | ||

G3 | Plan a trip by bus through several places in Toledo city, meeting some relatives | Variation of the problem | ||

4 | G1 | Draw three different star polygons inside a regular polygon of 37 edges | Variation of the problem | |

G2 | Draw a Eulerian cycle in a set of 5 points | Same problem already invented | ||

G3 | Draw a Eulerian cycle in this set of points using straight lines | Same problem already invented |

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**MDPI and ACS Style**

Blanco, R.; García-Moya, M.
Graph Theory for Primary School Students with High Skills in Mathematics. *Mathematics* **2021**, *9*, 1567.
https://doi.org/10.3390/math9131567

**AMA Style**

Blanco R, García-Moya M.
Graph Theory for Primary School Students with High Skills in Mathematics. *Mathematics*. 2021; 9(13):1567.
https://doi.org/10.3390/math9131567

**Chicago/Turabian Style**

Blanco, Rocío, and Melody García-Moya.
2021. "Graph Theory for Primary School Students with High Skills in Mathematics" *Mathematics* 9, no. 13: 1567.
https://doi.org/10.3390/math9131567