Reconstructibility of the $K_r$-count from $n-1$ cards

Abstract

The reconstruction conjecture by Kelly and Ulam was posed in the 40s and is still widely open today. A lemma by Kelly states that one can count subgraphs on at most n − 1 vertices given all the cards. We show that the clique count in G is reconstructable for all but one size of the clique from n − 1 cards. We extend this result by showing for graphs with average degree at most n/4 − 1 we can reconstruct the K r -count for all r, and that for r ≤ log n we can reconstruct the K r -count for every graph.