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

Abstract

The Reconstruction Conjecture of Kelly and Ulam states that any graph $G$ with $n\geq 3$ vertices can be reconstructed from the multiset $\mathcal{D}(G)$ of unlabelled subgraphs $G-v$ for all $v\in V(G)$. We refer to $\mathcal{D}(G)$ as the \emph{deck} of $G$ and $G-v\in \mathcal{D}(G)$ as the cards of $G$. This was posed in the 1940s and is still wide open today. In an effort to understand reconstructibility better, a growing collection of research is concerned with understanding what properties of $G$ can be reconstructed from a (potentially adversarially chosen) collection of $k$ cards for some $k< n$. In this paper, we show that the clique count of $G$ is reconstructible for all but one size of clique from any $n-1$ cards. We extend this result by showing that for graphs with average degree at most $3n/8-O(1)$ we can reconstruct the $K_r$-count for all $r$, and that for $r\le \log_2 n$ we can reconstruct the $K_r$-count for every graph on $n$ vertices.