Random Graph Isomorphism

  title={Random Graph Isomorphism},
  author={L{\'a}szl{\'o} Babai and Paul Erd{\"o}s and Stanley M. Selkow},
  journal={SIAM J. Comput.},
A straightforward linear time canonical labeling algorithm is shown to apply to almost all graphs (i.e. all but $o(2^{( \begin{subarray}{l} n \\ 2 \end{subarray} )} )$) of the $2^{( \begin{subarray}{l} n \\ 2 \end{subarray} )} $ graphs on n vertices). Hence, for almost all graphs X, any graph Y can be easily tested for isomorphism to X by an extremely naive linear time algorithm. This result is based on the following: In almost all graphs on n vertices, the largest $n^{0.15} $ degrees are… 
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Canonical labelling of graphs in linear average time
  • L. Babai, L. Kucera
  • Mathematics
    20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
  • 1979
It is proved that a simple vertex classification procedure results after only two refinement steps in a CL of random graphs with probability 1 - exp(-cn), and a linear time CL algorithm is obtained with only exp (-cn log n/log log n) probability of failure.
Degree sequences of random graphs
On the chromatic index of almost all graphs
BoLLOBás, Degree sequences of random graphs
  • Aarhus University,
  • 1978
Probabilistic analysis of a canonical numbering algorithm for graphs
  • Proc . Symposia in Pure Math
  • 1979
Probabilistic Methods in Combinatorics, Akadémiai Kiadó
  • Probabilistic Methods in Combinatorics, Akadémiai Kiadó
  • 1974
The beacon set approach to graph isomorphism
  • 1978
The particular corollary to Theorem 3 .6, that the vertex having maximum degree is unique in almost all graphs
    The fast approximate solution of hard combinatorial problems
    • Proc. 6th South-Eastern Conf . Combinatorics, Graph Theory and Computing (Florida Atlantic U
    • 1975