Ulam Numbers of Graphs

  • Stephen T. HedetniemiEmail author
Part of the Problem Books in Mathematics book series (PBM)


Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs having |V1| = |V2| and |E1| = |E2|. By an Ulam decomposition of order r we mean two partitions π1 = {E1,1, E1,2, …, E1,r} of E1 and π2 = {E2,1, E2,2, …, E2,r} of E2, having the properties for all 1 ≤ i ≤ r, (1) |E1,i| = |E2,i| and (2) the subgraph G[E1,i] induced by E1,i is isomorphic to the subgraph G[E2,i] induced by E2,i. In this note we generalize this concept, first introduced in 1979 by Chung et al. in Congr Numer 23:3–18, 1979.


  1. 1.
    F.R.K. Chung, P. Erdös, R.L. Graham, S.M. Ulam, F.F. Yao, Minimal decompositions of two graphs into pairwise isomorphic subgraphs. Congr. Numer. 23, 3–18 (1979). Proc. Tenth Southeastern Conf. on Combinatorics, Graph Theory and ComputingGoogle Scholar
  2. 2.
    R.L. Graham, Reflections on a theme of Ulam, in Graph Theory: Favorite Conjectures and Open Problems, vol. II (Springer, Springer, this volume)Google Scholar

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Authors and Affiliations

  1. 1.School of ComputingClemson UniversityClemsonUSA

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