Glasgow Mathematical Journal

Table of Contents - Volume 21 - Issue02

Research Article

Perfect codes in the graphs Ok and L(Ok)

D. H. Smitha1

a1 Polytechnic of Wales, Pontypridd, Wales

In [6] the question of the existence of perfect e-codes in the infinite family of distance-transitive graphs Ok was considered. It was pointed out that it is difficult to rule out completely any particular value of [6] because of the difficulty of working with the sphere packing condition. For e = 1, 2, 3 it can be seen from the results of [6] that the condition given by the generalisation of Lloyd's theorem is satisfied for infinitely many values of k. We shall show that this is not the case for e = 4 and we shall prove that there are no perfect 4-codes in Ok.

(Received April 26 1979)

REFERENCES

  • 1. Baker, A. and Davenport, H., The equations 3x2 - 2 = y2 and 8x2 - 7 = z2, Quart. J. Math. Oxford (2), 20 (1969), 129–137. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • 2. Biggs, N. L., Algebraic Graph Theory Cambridge Math. Tracts No. 67, (Cambridge University Press, London, 1974). [Google Scholar]
  • 3. Biggs, N. L., Perfect codes and distance-transitive graphs, in “Combinatorics” (Proceedings of the British Combinatorial Conference, 1973) (Cambridge University Press, London/New York 1974), 1–8. [Google Scholar]
  • 4. Biggs, N. L., Perfect codes in graphs, J. Combinatorial Theory, Ser. B 15 (1973), 289–296. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • 5. Hammond, P., q-coverings, codes and line graphs (to appear).
  • 6. Hammond, P. and Smith, D. H., Perfect codes in the graphs Ok. J. Combinatorial Theory, Ser. B 19 (1975), 239–255. [OpenURL Query Data]  [CrossRef]  [Google Scholar]