Minimal decompositions of graphs into mutually isomorphic subgraphs

Abstract

SupposeG n={G 1, ...,G k } is a collection of graphs, all havingn vertices ande edges. By aU-decomposition ofG n we mean a set of partitions of the edge setsE(G t ) of theG i , sayE(G t )==\(\sum\limits_{j = 1}^r {E_{ij} } \) E ij , such that for eachj, all theE ij , 1≦ik, are isomorphic as graphs. Define the functionU(G n) to be the least possible value ofr aU-decomposition ofG n can have. Finally, letU k (n) denote the largest possible valueU(G) can assume whereG ranges over all sets ofk graphs havingn vertices and the same (unspecified) number of edges.

In an earlier paper, the authors showed that

$$U_2 (n) = \frac{2}{3}n + o(n).$$

In this paper, the value ofU k (n) is investigated fork>2. It turns out rather unexpectedly that the leading term ofU k (n) does not depend onk. In particular we show

$$U_k (n) = \frac{3}{4}n + o_k (n),k \geqq 3.$$

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References

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    J. A. Bondy andU. S. R. Murty,Graph Theory with Applications, American Elsevier, New York (1976).

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    F. Frances Yao, Graph 2-isomorphism isNP-complete,Inf. Proc. Letters 9 (1979) 68–72.

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This work was performed while the author was a consultant at Bell Laboratoires.

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Chung, F.R.K., Erdős, P. & Graham, R.L. Minimal decompositions of graphs into mutually isomorphic subgraphs. Combinatorica 1, 13–24 (1981). https://doi.org/10.1007/BF02579173

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AMS subject classification (1980)

  • 05 C 35