plumiGraph theory: OCR A‑Level Further Maths questions
OCR A-Level Further Maths past-paper questions on 2.1 Graph theory, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 15 questions on Graph theory, as a list
- June 2025, Discrete Mathematics, Q2 (9 marks): (a) Graph G is shown below B A C F E D (i) Show that graph G is not bipartite. [2] (ii) Use Kuratowski’s Theorem to show that graph G is not planar. [3] (iii) D
- June 2025, AS Discrete Mathematics, Q5 (5 marks): Graph X is shown below. B D C A E H G F (a) Explain whether graph X is Eulerian, semi-Eulerian or neither. [1] (b) Give an example of a cycle on the graph above
- June 2024, Discrete Mathematics, Q1 (7 marks): At the end of each year the workers at an office take part in a gift exchange. Each worker randomly chooses the name of one other worker and buys a small gift f
- June 2024, AS Discrete Mathematics, Q1 (8 marks): There are six non-isomorphic trees with exactly six vertices. A student has drawn the diagram below showing six trees each with exactly six vertices. However, t
- June 2023, Discrete Mathematics, Q2 (8 marks): A graph is shown below. B C A D E F (a) Write down a cycle through all six vertices. [1] (b) Write down a continuous route that uses every arc exactly once. [2]
- June 2023, AS Discrete Mathematics, Q4 (10 marks): Graph G is a simply connected Eulerian graph with 4 vertices. (a) (i) Explain why graph G cannot be a complete graph. [2] (ii) Determine the number of arcs in g
- June 2022, Discrete Mathematics, Q1 (6 marks): Four children, A, B, C and D, discuss how many of the 23 birthday parties held by their classmates they had gone to. Each party was attended by at least one of
- June 2022, Discrete Mathematics, Q4 (13 marks): A connected graph is shown below. D G A F C B E H (a) Write down a path through exactly 7 of the vertices. [1] (b) Write down a cycle through exactly 6 of the v
- November 2021, Discrete Mathematics, Q2 (8 marks): A simply connected semi-Eulerian graph G has 6 vertices and 8 arcs. Two of the vertex degrees are 3 and 4. (a) (i) Determine the minimum possible vertex degree.
- November 2021, Discrete Mathematics, Q4 (13 marks): One of these graphs is isomorphic to K2,3. Graph A Graph B Graph C (a) Explain how you know that each of the other graphs is not isomorphic to K2,3. [2] The arc
- November 2020, Discrete Mathematics, Q1 (9 marks): This question is about the planar graph shown below. B E D G A C F (a) (i) Apply Kuratowski’s theorem to verify that the graph is planar. [2] (ii) Use Euler’s f
- November 2020, AS Discrete Mathematics, Q3 (8 marks): D B F H A G E C (a) By listing the vertices passed through, in the order they are used, give an example for the graph above of: (i) a walk from A to H that is n
- June 2019, Discrete Mathematics, Q1 (8 marks): Two graphs are shown below. A B P Q C D R S E T Graph G1 Graph G2 (a) (i) Prove that the graphs are isomorphic. [2] (ii) Verify that Euler’s formula holds for g
- June 2019, AS Discrete Mathematics, Q2 (10 marks): Two graphs are shown below. J K P Q L R M N S T Graph G1 Graph G2 (a) List the vertex degrees for each graph. [3] (b) Prove that the graphs are non‑isomorphic.
- June 2018, AS Discrete Mathematics, Q4 (8 marks): The complete bipartite graph K3, 4 connects the vertices {2, 4, 6} to the vertices {1, 3, 5, 7}. (i) How many arcs does the graph K3, 4 have? [1] (ii) Deduce ho
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