Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T09:24:14.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Isomorphic subgraphs having minimal intersections

Published online by Cambridge University Press:  09 April 2009

R. C. Mullin ,
B. K. Roy  and
P. J. Schellenberg
R. C. Mullin
Affiliation:
University of Waterloo Waterloo, Ontario, Canada
B. K. Roy
Affiliation:
University of Waterloo Waterloo, Ontario, Canada
P. J. Schellenberg
Affiliation:
University of Waterloo Waterloo, Ontario, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a finite graph H and G, a subgraph of it, we define σ (G, H) to be the largest integer such that every pair of subgraphs of H, both isomorphic to G, has at least σ(G, H) edges in common; furthermore, R(G, H) is defined to be the maximum number of subgraphs of H, all isomorphic to G, such that any two of them have σ(G, H) edges common between them. We are interested in the values of σ(G, H) and R(G, H) for general H and G. A number of combinatorial problems can be considered as special cases of this question; for example, the classical set-packing problem is equivalent to evaluating R (G, H) where G is a complete subgraph of the complete graph H and σ(G, H) = 0, and the decomposition of H into subgraphs isomorphic to G is equivalent to showing that σ(G, H) = 0 and R(G, H) = ε(H)/ε(G) where ε(H), ε(G) are the number of edges in H, G respectively.

A result of S. M. Johnson (1962) gives an upper bound for R(G, H) in terms of σ(G, H). As a corollary of Johnson's result, we obtain the upper bound of McCarthy and van Rees (1977) for the Cordes problem. The remainder of the paper is a study of σ (G, H) and R(G, H) for special classes of graphs; in particular, H is a complete graph and G is, in most instances, a union of disjoint complete subgraphs.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 3 , December 1983 , pp. 287 - 306
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Bondy, J. A. and Murty, U. S. R. (1976), Graph theory with applications (Macmillan Press, Hong Kong).CrossRefGoogle Scholar
[2]Bose, R. C. (1947), ‘On resolvable series of balanced incomplete block designs’, Sankhya 8, 251–257.Google Scholar
[3]Cordes, C. M. (1978), ‘A new type of combinatorial design’, J. Combinatorial Theory, Ser. A. 24, 251–257.CrossRefGoogle Scholar
[4]Hartman, A., Mullin, R. C. and Stinson, D. R. (1980), Exact covering configurations and Steiner systems, Research Report CORR 80–40 (Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada).Google Scholar
[5]Hall, M. Jr, (1967), Combinatorial theory (Blaisdell Publishing Co., Wallham, Mass).Google Scholar
[6]Johnson, S. M. (1962), ‘A new upper bound for error-correcting codes’, IRE Trans. Informat. Theory 8, 203–207.CrossRefGoogle Scholar
[7]McCarthy, D. and van Rees, G. H. J. (1977), ‘Some results on a combinatorial problem of Cordes’, J. Austral. Math. Soc. Ser. A. 23, 439–452.CrossRefGoogle Scholar
[8]Mullin, R. C. and Stanton, R. G. (1978), ‘A characterization of pseudo-affine designs and their relation to a problem of Cordes’, Ann. Discrete Math. 2, 231–238.CrossRefGoogle Scholar
[9]Nemeth, E. (1976), ‘A note on a minimax problem in scheduling’, Proceedings 6th Manitoba Conference on Numerical Math., pp. 343–349 (Winnipeg, Manitoba).Google Scholar
[10]Ryser, H. J. (1950), ‘A note on a combinatorial problem’, Proc. Amer. Math. Soc. 1, 422–424.CrossRefGoogle Scholar
[11]Wallis, W. D., Street, A. P. and Wallis, J. S. (1972), Combinatorics: Room squares, sum-free sets, Hadamard matrices, Lecture Notes in Mathematics, no. 292 (Springer-Verlag, Berlin).CrossRefGoogle Scholar