Some History on the Reconstruction Conjecture Allen Schwenk January 7, 2012 1 Some Recollections on the Reconstruction Conjecture Allen Schwenk January 7, 2012 2 Some Anecdotes on the Reconstruction Conjecture Allen Schwenk January 7, 2012 3 Some Gossip on the Reconstruction Conjecture Allen Schwenk January 7, 2012 4 What is the original form of the Reconstruction Conjecture? 5 What is the original form of the Reconstruction Conjecture? Two graphs are called hypomorphic if there is a bijection α:V(G) →V(H ) of the vertices such that the vertex-deleted subgraphs are isomorphic in pairs: € 6 What is the original form of the Reconstruction Conjecture? Two graphs are called hypomorphic if there is a bijection α:V(G) →V(H ) of the vertices such that the vertex-deleted subgraphs are isomorphic in pairs: Hypomorphic: For each i, 1 ≤ i ≤ n, G − v ≅ H −α(v ) . i i € € 7 What is the original form of the Reconstruction Conjecture? Two graphs are called hypomorphic if there is a bijection α:V(G) →V(H ) of the vertices such that the vertex-deleted subgraphs are isomorphic in pairs: Hypomorphic: For each i, 1 ≤ i ≤ n, G − v ≅ H −α(v ) . i i € Reconstruction Conjecture: also called the RC: € 8 What is the original form of the Reconstruction Conjecture? Two graphs are called hypomorphic if there is a bijection α:V(G) →V(H ) of the vertices such that the vertex-deleted subgraphs are isomorphic in pairs: Hypomorphic: For each i, 1 ≤ i ≤ n, G − v ≅ H −α(v ) . i i € Reconstruction Conjecture: also called the RC: € If G and H are hypomorphic, then G and H must be isomorphic. 9 This is the form popularized by Stan Ulam in 1960, A collection of mathematical puzzles. 10