Lesson DA1-DA3

DA1-DA3 Graph language, Eulerian graphs and Euler's formula Quiz: AQA Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DA1-DA3, Graph language, Eulerian graphs and Euler's formula: 20 multiple choice questions for the AQA Further Maths (7367), Unit 5: Optional application 3: discrete mathematics, written with Revision Ninja.

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The 20 questions

  1. Which definition describes an Eulerian graph?

    • A connected graph in which every vertex has even degree
    • A graph containing a cycle that passes through every vertex exactly once
    • A connected graph in which every vertex has odd degree
    • A graph with exactly two vertices of odd degree

  2. Which definition describes a semi-Eulerian graph?

    • A graph with exactly two vertices of degree 1
    • A connected graph in which every vertex has even degree
    • A connected graph with exactly two vertices of even degree
    • A connected graph with exactly two vertices of odd degree

  3. What is a Hamiltonian cycle in a graph?

    • A trail that uses every edge of the graph exactly once
    • A closed walk that visits every edge at least twice
    • A cycle that uses every edge of the graph exactly once
    • A cycle that passes through every vertex of the graph exactly once

  4. Which statement is Euler's formula for a connected planar graph with v vertices, e edges and f faces (including the outer face)?

    • v - e + f = 2
    • v + e - f = 2
    • v - e + f = 1
    • e - v + f = 2

  5. A connected planar graph has 6 vertices and 9 edges. How many faces does it have, including the outer face?

    • 3
    • 5
    • 4
    • 6

  6. The vertices of a simple graph have degrees 3, 3, 2, 2 and 4. How many edges does the graph have?

    • 6
    • 14
    • 7
    • 5

  7. A connected graph has vertices of degrees 2, 4, 4, 3 and 5. Which statement is correct?

    • It is neither Eulerian nor semi-Eulerian, since it has more than one odd vertex
    • It is Eulerian, since the degree sum 18 is even
    • It is Eulerian, since every vertex has degree at least 2
    • It is semi-Eulerian but not Eulerian, since exactly two vertices have odd degree

  8. A complete graph on five vertices, in which every vertex has degree 4, has how many edges?

    • 8
    • 10
    • 5
    • 20

  9. The complete graph K4 is drawn in the plane with no crossings. How many faces does this plane drawing have?

    • 2
    • 6
    • 3
    • 4

  10. A connected planar graph has 10 edges and 6 faces. How many vertices does it have?

    • 4
    • 18
    • 8
    • 6

  11. A connected graph has vertices of degrees 2, 2, 4, 4 and 4. Which statement is correct?

    • It is not Eulerian, since its degrees are not all equal
    • It is Eulerian, since every vertex has even degree and the graph is connected
    • It is Eulerian only if it is also planar
    • It is semi-Eulerian, since it has five vertices

  12. Which operation on a graph inserts a new vertex of degree 2 in the middle of an edge, while keeping the graph's essential structure?

    • Removing a vertex together with all its incident edges
    • Subdivision of an edge
    • Merging two vertices into a single vertex of degree 4
    • Adding an extra edge between two existing vertices

  13. In graph theory, what is a loop and what is a multiple edge?

    • A loop joins two vertices of degree 1, and a multiple edge is a path of length two
    • A loop is an edge with weight zero, and a multiple edge has no endpoints
    • A loop joins a vertex to itself, and a multiple edge is one of two or more edges joining the same pair of vertices
    • A multiple edge joins a vertex to itself, and a loop joins two different vertices

  14. What is the difference between a trail and a path in a graph?

    • A trail is a walk with no repeated edges, while a path has no repeated vertices
    • A trail has no repeated vertices, while a path has no repeated edges
    • A trail must be closed, while a path must be open
    • A trail and a path are both walks that may repeat vertices and edges

  15. A vertex of a graph has one loop and two other edges. What is the degree of the vertex?

    • 2
    • 3
    • 1
    • 4

  16. Which statement about the triangle K3, the cycle on three vertices, is correct?

    • It is neither Eulerian nor Hamiltonian, since its vertices do not all have the same degree
    • It is Eulerian but not Hamiltonian, since it has only three vertices
    • It is both Eulerian and Hamiltonian, since every vertex has degree 2 and it is a cycle
    • It is Hamiltonian but not Eulerian, since three is an odd number of vertices

  17. A connected planar graph has 9 vertices, all of degree 4. How many faces does it have, including the outer face?

    • 10
    • 11
    • 12
    • 9

  18. A connected graph has exactly four vertices of odd degree. Which statement is correct?

    • It is not Eulerian, and it needs at least two trails to cover all its edges
    • It needs exactly four separate trails to cover all of its edges
    • It is semi-Eulerian, since it has an even number of odd vertices
    • It is Eulerian, since the number of odd vertices is even

  19. Which statement about the relationship between Eulerian and Hamiltonian properties is correct?

    • Every Eulerian graph is necessarily Hamiltonian
    • A graph can be Eulerian without being Hamiltonian, and can be Hamiltonian without being Eulerian
    • A graph is Hamiltonian if and only if it is connected
    • Every Hamiltonian graph is necessarily Eulerian

  20. A connected planar graph has 7 vertices and 5 faces. How many edges does it have?

    • 9
    • 14
    • 12
    • 10

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