Ulam Numbers of Graphs

Abstract

Let G ₁ = (V ₁, E ₁) and G ₂ = (V ₂, E ₂) be two graphs having |V ₁| = |V ₂| and |E ₁| = |E ₂|. By an Ulam decomposition of order r we mean two partitions π ₁ = {E _1,1, E _1,2, …, E _1,r} of E ₁ and π ₂ = {E _2,1, E _2,2, …, E _2,r} of E ₂, having the properties for all 1 ≤ i ≤ r, (1) |E _1,i| = |E _2,i| and (2) the subgraph G[E _1,i] induced by E _1,i is isomorphic to the subgraph G[E _2,i] induced by E _2,i. In this note we generalize this concept, first introduced in 1979 by Chung et al. in Congr Numer 23:3–18, 1979.