Lesson 7.02g-p

7.02g-p Eulerian and Hamiltonian graphs, isomorphism, digraphs, planarity and networks Quiz: OCR Further Maths, Unit 4

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson 7.02g-p, Eulerian and Hamiltonian graphs, isomorphism, digraphs, planarity and networks: 20 multiple choice questions for the OCR Further Maths (H245), Unit 4: Discrete Mathematics (Y544), written with Revision Ninja.

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The 20 questions

  1. Which graph condition guarantees the presence of an Eulerian circuit?

    • Any vertex odd
    • All degrees even
    • Maximum degree two
    • All degrees odd

  2. How many odd-degree vertices does a semi-Eulerian connected graph have?

    • At least four
    • Exactly one
    • Exactly two
    • Exactly zero

  3. What does a Hamiltonian cycle in a graph visit exactly once?

    • Every component
    • Every face
    • Every vertex
    • Every edge

  4. A connected planar graph has 6 vertices and 9 edges. How many faces does it have?

    • 6
    • 4
    • 3
    • 5

  5. Which non-planar complete bipartite graph features in Kuratowski's theorem?

    • K4,4
    • K3,3
    • K2,3
    • K3,4

  6. A graph has 8 edges. What is the sum of the degrees of all its vertices?

    • 16
    • 24
    • 32
    • 8

  7. What term describes two graphs that have identical structural connectivity?

    • Eulerian
    • Isomorphic
    • Planar
    • Bipartite

  8. How many edges does the complete graph K6 contain?

    • 12
    • 15
    • 30
    • 18

  9. In any directed graph, how does total in-degree compare to total out-degree?

    • Out-degree is greater
    • Always zero
    • In-degree is greater
    • They are equal

  10. What is the maximum number of edges a simple connected planar graph with 5 vertices can have?

    • 9
    • 7
    • 8
    • 10

  11. In discrete mathematics, what defines a network compared to a standard graph?

    • A weighted graph
    • A directed tree
    • A planar graph
    • A complete graph

  12. A digraph has vertices with out-degrees 2, 3, 1, and 4. How many edges does it have?

    • 5
    • 20
    • 8
    • 10

  13. Which complete graph is the smallest non-planar complete graph according to Kuratowski's theorem?

    • K4
    • K6
    • K3
    • K5

  14. If a connected graph has a trail starting and ending at distinct vertices, it is called:

    • Semi-Eulerian
    • Hamiltonian
    • Eulerian
    • Semi-Hamiltonian

  15. A connected planar graph has 4 faces, each bounded by 3 edges. How many edges are there?

    • 12
    • 4
    • 6
    • 8

  16. What is the maximum number of edges in a simple planar bipartite graph with 6 vertices?

    • 6
    • 8
    • 12
    • 10

  17. Which property must always be identical in two isomorphic simple graphs?

    • Degree sequence
    • Edge weights
    • Vertex labels
    • Spatial layout

  18. How many edges are in the complete bipartite graph K3,4?

    • 10
    • 14
    • 7
    • 12

  19. By Dirac's theorem, a simple graph with n vertices is Hamiltonian if every degree is at least:

    • 2n
    • n / 3
    • n / 2
    • n - 1

  20. What key property defines a planar graph when drawn on a flat surface?

    • All degrees even
    • Fully connected
    • No cycles exist
    • No edges cross

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