Lesson DA4-DA5

DA4-DA5 Planarity, complete and bipartite graphs Quiz: AQA Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DA4-DA5, Planarity, complete and bipartite graphs: 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 statement is Kuratowski's Theorem?

    • A graph is planar if and only if it contains no cycle of length greater than four
    • A graph is planar if and only if it has no Eulerian circuit
    • A graph is planar if and only if every vertex has degree at most four
    • A graph is planar if and only if it contains no subdivision of K5 or K3,3

  2. What is a complete graph Kn?

    • A simple graph in which every pair of distinct vertices is joined by exactly one edge
    • A graph in which every vertex has degree 2
    • A connected graph with no cycles
    • A graph in which each vertex is joined to exactly half of the other vertices

  3. What is a bipartite graph?

    • A graph with exactly two vertices of odd degree
    • A graph with exactly two connected components
    • A graph whose vertices can be split into two disjoint sets, with every edge joining a vertex in one set to a vertex in the other
    • A graph whose vertices can be split into two sets, with every edge lying inside one of the sets

  4. What is the complement of a simple graph G with the same vertex set?

    • The graph with the same edges as G but with its vertices relabelled
    • The graph formed by reversing the direction of every edge of G
    • The graph obtained by deleting all the edges of G
    • The graph whose edges are exactly the pairs of vertices that are not joined by an edge in G

  5. How many edges does the complete graph K7 have?

    • 49
    • 21
    • 42
    • 14

  6. How many edges does the complete bipartite graph K3,4 have?

    • 6
    • 12
    • 7
    • 24

  7. The complete bipartite graph K3,3 has 6 vertices. How many edges does its complement have?

    • 3
    • 12
    • 9
    • 6

  8. What is the maximum number of edges a simple planar graph with 6 vertices can have?

    • 9
    • 10
    • 12
    • 15

  9. What is the maximum number of edges a simple bipartite planar graph with 6 vertices can have?

    • 6
    • 9
    • 8
    • 12

  10. Which of the following graphs is non-planar according to Kuratowski's Theorem?

    • The cycle on six vertices
    • K3,3
    • K4
    • K2,3

  11. A simple graph has 5 edges. What is the sum of all the entries in its adjacency matrix?

    • 25
    • 20
    • 5
    • 10

  12. In the adjacency matrix of a simple graph, what does the sum of the entries in the row for a vertex give?

    • The number of cycles passing through that vertex
    • The number of vertices at distance two from that vertex
    • The number of edges in the whole graph
    • The degree of that vertex

  13. Is a cycle with an odd number of vertices bipartite?

    • No, because its vertices cannot be split into two sets with every edge joining the two sets
    • Yes, but only when the cycle has more than five vertices
    • Yes, because every cycle is bipartite
    • Yes, because odd cycles are always planar

  14. Can K5 be shown to be non-planar using the bound e ≤ 3v - 6 for simple planar graphs?

    • No, because 10 is less than 3 × 5 = 15
    • Yes, because K5 has 10 edges, which exceeds 3 × 5 - 6 = 9
    • No, because the bound applies only to graphs that contain a triangle
    • Yes, because K5 has 5 edges, which exceeds 3 × 5 - 6 = 9

  15. Is the complete bipartite graph K2,3 planar?

    • Yes, because it contains no subdivision of K5 or K3,3
    • Yes, but only if one of its edges is drawn with a crossing
    • No, because it is bipartite
    • No, because it has six edges

  16. For which smallest value of n does the complete graph Kn break the planar edge bound e ≤ 3n - 6?

    • 3
    • 4
    • 6
    • 5

  17. A simple connected graph has 7 vertices and 16 edges. Which statement is correct?

    • It is planar, since every graph with fewer than 20 edges is planar
    • It is not planar, since 16 edges exceed the planar bound 3 × 7 - 6 = 15
    • It is not planar, since every graph with 7 vertices contains a subdivision of K5
    • It is planar, since 16 is less than 7 squared

  18. Could a simple bipartite graph with 10 vertices and 20 edges be planar?

    • Yes, because all bipartite graphs are planar
    • Yes, because 20 is less than 3 × 10 - 6 = 24
    • No, because a simple planar bipartite graph on 10 vertices has at most 2 × 10 - 4 = 16 edges
    • No, because bipartite graphs contain no cycles

  19. Which statement about Kuratowski's Theorem and planarity is correct?

    • Any graph containing a cycle of length five is non-planar
    • A graph is non-planar if it contains a subdivision of K5 or K3,3, and a planar graph contains neither
    • K5 and K3,3 are planar graphs, but only after one edge has been removed
    • A graph is non-planar if it has more than 3v - 6 edges, and this edge count is the only test needed

  20. How many entries equal 1 in the adjacency matrix of the complete graph K4?

    • 4
    • 16
    • 12
    • 6

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