First stage comprehensive exams

The first stage comprehensive exams are written exams that will be offered once a year and must be attempted within four terms of the first registration. The student must write one exam from two of the following three caegories:

Category one Category two Category three
  • Combinatorial enumeration
  • Graph theory
  • Continuous optimization
  • Discrete optimization
  • Cryptography
  • Quantum computing

The Choice of the exams is made by the student, in consultation with their supervisor.

The first-stage comprehensive examination requirement is satisfied by passing both exams.  If you do not pass, you may be asked fulfil other requirements or to retake both exams.  Students who fail a retake will be required withdraw from the program.

The comprehensive exam asses the student's grasp of the core background of their area as well as the student's ability to write formal mathematics.

The syllabus for the exams is listed below.

Combinatorial enumeration suggested references

Graph theory suggested references

  • Graph Theory by R. Diestel, Second Edition Graduate Texts in Mathematics 173 Springer, New York, 2000.
    • Matchings (2.1, 2.2)
    • Connectivity (3.1, 3.2, 3.3)
    • Planarity (4.1-4.4)
    • Colourings (5.1, 5.2, 5.3)
    • Flows (6.3, 6.4, 6.6)
    • Turán's Theorem (7.1)
    • Ramsey's Theorem (9.1)
    • Hamiltonicity (10.1, 10.2)
    • PLUS the problems associated with the above sections
  • Algebraic Graph Theory by C. Godsil and G. Royle, Graduate Texts in Mathematics 207, Springer, New York, 2001.
    • Transitivity (3)
    • Matrices and Graphs (8.1-8.8)
    • Interlacing (9.1, 9.2, 9.3)
    • Strongly Regular Graphs (10.1-10.5)
  • Introduction to Graph Theory by D.B. West, Prentice Hall, Upper Saddle River, NJ 1996.
    • Connectivity (p.136-138, 4.2)
    • Ramsey's Theorem (8.3)
  • Introduction to Graph Theory by R.J. Wilson, Third Edition Longman, Essex, 1985.

        Directed Graphs (7)

Continuous optimization suggested references

Discrete optimization suggested references

Quantum computing suggested references

Cryptography suggested references

Old first stage comprehensive exams