The double reconstruction conjecture about finite colored hypergraphs (Q912131)

One of the most famous open problems about simple graphs is the Ulam's Reconstruction Conjecture which asserts any two hypomorphic simple graphs of order greater than two are isomorphic. \textit{W. L. Kocay} [J. Comb. Theory, Ser. B 42, 46-63 (1987; Zbl 0578.05052)] presented an infinite family of pairs \((X_ n,Y_ n)\), \(n\geq 2\), of 3-hypergraphs which is a hypermorphic pair but not an isomorphic pair. This paper introduces a new reconstruction conjecture about colored hypergraphs called the Double Reconstruction Conjecture. It is shown among others that (1) the restricted version of this conjecture about simple graphs is equivalent to the Ulam's Conjecture, and (2) each pair, \((X_ n,Y_ n)\), \(n\geq 2\), above does not satisfy the condition in this conjecture.