The double reconstruction conjectures about colored hypergraphs and colored directed graphs

Abstract

This paper introduces two new reconstruction conjectures about colored hypergraphs and colored directed graphs, and presents results about these two conjectures some of which are as follows: (1) the restricted version of the first conjecture to simple graphs is equivalent to the Ulam's Conjecture; (2) Kokay's each hypomorphic pair (X n ,Y n ), n=2,3,..., of 3-hypergraphs does not satisfy the conditions in the first conjecture; (3) any two (0, k)-hypergraphs G and G′ of order n are isomorphic if there exists a bijection α: V→V′ and an integer m, k ≤ m ≤ n-k, such that for any k-subset W of V(G), G- W is isomorphic to G′- α(W); (4) the validity of the second conjecture implies the validity of the Ulam's Conjecture; (5) Stockmeyer's hypomorphic tournaments, B n and C n , n ≤ 1, are not doubly m-hypomorphic for any 2n−2+1≤m≤ 3 · 2n−2.